A010054: a(n) = 1 if n is a triangular number, otherwise 0.

OEIS matches ?

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OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1;
  4: 1;
  5: 1;
  6: 1;
  7: 1;
  8: 1;
  9: 1;

The row polynomials satisfy P(n,x) = P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-01080164ff17b795, e-17b9246d5ae52c95, e-249b7eaa76a03e97, e-35224690d27a26b2, e-3c3f8de32007e148, e-3e8a15c56023eda3, e-4b431952cda59e9d, e-68bece1a3264cb39, e-99f76b7466efc63d, e-a4c5b7fb551c21c9, e-a877a82e81f28b82, e-abb1d38cae3c521a, e-b6c6a979053116b3, e-ccf167b9b4684400, e-e9fed022b38b6e62
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-01080164ff17b795, e-17b9246d5ae52c95, e-249b7eaa76a03e97, e-35224690d27a26b2, e-3c3f8de32007e148, e-3e8a15c56023eda3, e-4b431952cda59e9d, e-54652e8af65208a6, e-68bece1a3264cb39, e-8ef103093ba4177d, e-99f76b7466efc63d, e-a4c5b7fb551c21c9, e-a877a82e81f28b82, e-abb1d38cae3c521a, e-b6c6a979053116b3, e-b80fd10f36f0471f, e-ccf167b9b4684400, e-e9fed022b38b6e62, e-f5eca334b4637498
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-01080164ff17b795, e-17b9246d5ae52c95, e-249b7eaa76a03e97, e-35224690d27a26b2, e-3c3f8de32007e148, e-3e8a15c56023eda3, e-4b431952cda59e9d, e-54652e8af65208a6, e-68bece1a3264cb39, e-8ef103093ba4177d, e-99f76b7466efc63d, e-a4c5b7fb551c21c9, e-a877a82e81f28b82, e-abb1d38cae3c521a, e-b6c6a979053116b3, e-b80fd10f36f0471f, e-ccf167b9b4684400, e-e9fed022b38b6e62, e-f5eca334b4637498
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: T(p) = p(a) g; an input with every root <= a has p(a) of the sign of its leading coefficient
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-05b27694ad6bb7e9
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-695286cd251b7e4d, e-88f7d5697e008187
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-695286cd251b7e4d, e-88f7d5697e008187
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-695286cd251b7e4d, e-88f7d5697e008187
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f5eca334b4637498
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-b80fd10f36f0471f
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1564556493/268435456, -46056243/268435456]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-8ef103093ba4177d
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-54652e8af65208a6

Proof chains

22 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A008277 e-17b9246d5ae52c95

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 0 e-01080164ff17b795

      Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

      • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-249b7eaa76a03e97 root: theorem check

          Parent-row provers of polytool polytool ?

        • basis_map A013609 e-35224690d27a26b2

          Geometric basis map (elementary) polytool ?

          • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
            • gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148

              Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

              • premise (sequence): Even-numbered rows of Pascal's triangle.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • parent_provers periodic_product e-0029b53144d984cf root: theorem check

                  Parent-row provers of polytool polytool ?

                • preserver_image reversal, shift 0 e-a3e5463f21dc8d9d

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): Even-numbered rows of Pascal's triangle.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ parent_provers periodic_product e-0029b53144d984cf (expanded above)
                    • basis_map A007318 e-fc612a08cfe0bbb6

                      Geometric basis map (elementary) polytool ?

                      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

                          Parent-row provers of polytool polytool ?

                        • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

                          Affine, reversal and Möbius substitutions (elementary) polytool ?

                          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                            • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
                    • preserver_image affine, shift 0 e-5b79b5b4d4f1c577

                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
                        • ↵ preserver_image mobius, shift 0 e-12303ad0e0b9f8eb (expanded above)
                • ↵ basis_map A007318 e-fc612a08cfe0bbb6 (expanded above)
                • ↵ preserver_image affine, shift 0 e-5b79b5b4d4f1c577 (expanded above)
            • basis_map A013610 e-3e8a15c56023eda3

              Geometric basis map (elementary) polytool ?

              • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                • preserver_image reversal, shift 0 e-4b431952cda59e9d

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d

                      Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?

                      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
                        • ↵ preserver_image mobius, shift 0 e-12303ad0e0b9f8eb (expanded above)
                    • parent_provers periodic_product e-b7eca039219a7f53 root: theorem check

                      Parent-row provers of polytool polytool ?

                    • preserver_image mobius, shift 0 e-7e44b66e04c497d6

                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                      • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                        • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                • preserver_image reversal, shift 0 e-68bece1a3264cb39

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
                    • ↵ preserver_image mobius, shift 0 e-12303ad0e0b9f8eb (expanded above)
                • basis_map A007318 e-99f76b7466efc63d

                  Geometric basis map (elementary) polytool ?

                  • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                    • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                    • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                    • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                    • veronese_section r=2, j=0, shift 0 e-a4c5b7fb551c21c9

                      Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?

                      • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                        • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                        • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                        • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                        • preserver_image reversal, shift 0 e-a877a82e81f28b82

                          Affine, reversal and Möbius substitutions (elementary) polytool ?

                          • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                            • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                            • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                            • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                            • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                            • gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a

                              Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

                              • premise (sequence): binomial
                                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                • theorem_check liu_wang e-317eb4ccca60c6ac root: theorem check

                                  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

                                • preserver_image reversal, shift 0 e-033cf842aad3bdf7

                                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                                  • premise (sequence): binomial
                                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                    • ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
                                    • basis_map A007318 e-095baa2a9dfe188f

                                      Geometric basis map (elementary) polytool ?

                                      • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                        • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                                        • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                                        • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
                                    • parent_provers periodic_product e-13983862a411779c root: theorem check

                                      Parent-row provers of polytool polytool ?

                                    • preserver_image reversal, shift 0 e-989c95e5c893fce9

                                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                                      • premise (sequence): Riordan array (1-x,x/(1-x)).
                                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                        • parent_provers periodic_product e-b62074ba45057e82 root: theorem check

                                          Parent-row provers of polytool polytool ?

                                    • gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea

                                      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

                                      • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
                                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                        • preserver_image reversal, shift 0 e-0e058c428dfdfeb6

                                          Affine, reversal and Möbius substitutions (elementary) polytool ?

                                          • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
                                            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                            • parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check

                                              Parent-row provers of polytool polytool ?

                                        • ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)
                                    • preserver_image affine, shift 0 e-3a52de7742ed9314

                                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                                      • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                        • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                                        • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                                        • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
                                • ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
                                • ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
                                • ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
                                • ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
                                • ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
                            • preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3

                              Laguerre’s theorem on differential operators [SP14] polytool ?

                              • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                                • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                                • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                                • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                                • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                                • preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400

                                  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

                                  • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                    • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                                    • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                                    • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                                    • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                                    • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                                    • basis_map A013611 e-e9fed022b38b6e62

                                      Geometric basis map (elementary) polytool ?

                                      • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                        • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                                        • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                                        • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                                        • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                                        • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                                        • preserver_image affine, shift 0 e-f5eca334b4637498

                                          Affine, reversal and Möbius substitutions (elementary) polytool ?

                                          • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                                            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                            • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                                            • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                                            • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                                            • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                                            • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                                            • preserver_image mobius, shift 0 e-b80fd10f36f0471f

                                              Affine, reversal and Möbius substitutions (elementary) polytool ?

                                              • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                                                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                                                • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
                                                • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
                                                • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
                                                • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
                                                • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                                                • basis_map A008288 e-8ef103093ba4177d

                                                  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

                                                  • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                                                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [-10368968295/4294967296, 0]
                                                    no recorded property covers this requirement
                                                • basis_map A008288 e-54652e8af65208a6

                                                  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

                                                  • premise (sequence): a(n) = 1 if n is a triangular number, otherwise 0.
                                                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [-10368968295/4294967296, 0]
                                                    no recorded property covers this requirement
                                            • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                                            • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
                                        • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
                                        • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                                        • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
                                    • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
                                    • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
                                    • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                                    • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
                                • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
                                • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
                                • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
                                • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                                • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
                            • ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
                            • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
                            • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
                            • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
                            • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                            • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
                        • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                        • ↵ preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3 (expanded above)
                        • ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
                        • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
                        • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
                        • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
                        • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                        • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
                    • ↵ preserver_image reversal, shift 0 e-a877a82e81f28b82 (expanded above)
                    • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                    • ↵ preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3 (expanded above)
                    • ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
                    • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
                    • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
                    • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
                    • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                    • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
                • ↵ veronese_section r=2, j=0, shift 0 e-a4c5b7fb551c21c9 (expanded above)
                • ↵ preserver_image reversal, shift 0 e-a877a82e81f28b82 (expanded above)
                • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
                • ↵ preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3 (expanded above)
                • ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
                • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
                • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
                • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
                • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
                • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
            • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
            • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
            • ↵ basis_map A007318 e-99f76b7466efc63d (expanded above)
            • ↵ veronese_section r=2, j=0, shift 0 e-a4c5b7fb551c21c9 (expanded above)
            • ↵ preserver_image reversal, shift 0 e-a877a82e81f28b82 (expanded above)
            • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
            • ↵ preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3 (expanded above)
            • ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
            • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
            • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
            • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
            • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
        • ↵ basis_map A013610 e-3e8a15c56023eda3 (expanded above)
        • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
        • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
        • ↵ basis_map A007318 e-99f76b7466efc63d (expanded above)
        • ↵ veronese_section r=2, j=0, shift 0 e-a4c5b7fb551c21c9 (expanded above)
        • ↵ preserver_image reversal, shift 0 e-a877a82e81f28b82 (expanded above)
        • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
        • ↵ preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3 (expanded above)
        • ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
        • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
        • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
        • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
        • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
        • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
    • ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
    • ↵ basis_map A013609 e-35224690d27a26b2 (expanded above)
    • ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
    • ↵ basis_map A013610 e-3e8a15c56023eda3 (expanded above)
    • ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
    • ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
    • ↵ basis_map A007318 e-99f76b7466efc63d (expanded above)
    • ↵ veronese_section r=2, j=0, shift 0 e-a4c5b7fb551c21c9 (expanded above)
    • ↵ preserver_image reversal, shift 0 e-a877a82e81f28b82 (expanded above)
    • ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
    • ↵ preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3 (expanded above)
    • ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
    • ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
    • ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
    • ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
    • ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
    • ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
• ↵ basis_map A013609 e-35224690d27a26b2 (expanded above)
• ↵ basis_map A013610 e-3e8a15c56023eda3 (expanded above)
• ↵ basis_map A008288 e-54652e8af65208a6 (expanded above)
• ↵ basis_map A008288 e-8ef103093ba4177d (expanded above)
• ↵ basis_map A007318 e-99f76b7466efc63d (expanded above)
• ↵ basis_map A013611 e-e9fed022b38b6e62 (expanded above)
• basis_map_theorem e-05b27694ad6bb7e9 root: theorem check

  Geometric basis map (elementary) polytool ?

• ↵ gamma_polynomial scale 1, shift 0 e-3c3f8de32007e148 (expanded above)
• ↵ gamma_polynomial scale 1, shift 0 e-abb1d38cae3c521a (expanded above)
• ↵ parent_provers periodic_product e-249b7eaa76a03e97 (expanded above)
• ↵ preserver_image euler_polynomial, shift 0 e-01080164ff17b795 (expanded above)
• ↵ preserver_image reversal, shift 0 e-4b431952cda59e9d (expanded above)
• ↵ preserver_image reversal, shift 0 e-68bece1a3264cb39 (expanded above)
• preserver_image reversal, shift 0 e-695286cd251b7e4d

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): basis-map coefficients of A scaled Hermite triangle.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-470643e70af29063 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-f9cce4eb6656235e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of A scaled Hermite triangle.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-470643e70af29063 (expanded above)
• preserver_image reversal, shift 0 e-88f7d5697e008187

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-ad67990681a86223

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)
• ↵ preserver_image reversal, shift 0 e-a877a82e81f28b82 (expanded above)
• ↵ preserver_image derivative_polynomial, shift 0 e-b6c6a979053116b3 (expanded above)
• ↵ preserver_image mobius, shift 0 e-b80fd10f36f0471f (expanded above)
• ↵ preserver_image euler_polynomial, shift 0 e-ccf167b9b4684400 (expanded above)
• ↵ preserver_image affine, shift 0 e-f5eca334b4637498 (expanded above)
• ↵ veronese_section r=2, j=0, shift 0 e-a4c5b7fb551c21c9 (expanded above)

Used in 19 proof steps

• e-17b9246d5ae52c95 basis_map (A008277) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-35224690d27a26b2 basis_map (A013609) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-3e8a15c56023eda3 basis_map (A013610) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-54652e8af65208a6 basis_map (A008288) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-8ef103093ba4177d basis_map (A008288) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-99f76b7466efc63d basis_map (A007318) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-e9fed022b38b6e62 basis_map (A013611) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-01080164ff17b795 preserver_image (euler_polynomial, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-74abd0d742dd5162 preserver_image (reversal, shift 0) → Triangle read by rows: row n consists of n zeros followed by n+1.
• e-941e1ded9756e612 preserver_image (derivative_of_power_multiple, shift 0) → Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• e-9c0dd75b125bab92 preserver_image (reversal, shift 0) → Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• e-a877a82e81f28b82 preserver_image (reversal, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-b6c6a979053116b3 preserver_image (derivative_polynomial, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-b80fd10f36f0471f preserver_image (mobius, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-ccf167b9b4684400 preserver_image (euler_polynomial, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-d2ec3b49898958f3 preserver_image (reversal, shift 0) → Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• e-e8da6a4618f02a9d preserver_image (reversal, shift 0) → 3n + 1 preceded by n zeros.
• e-f5eca334b4637498 preserver_image (affine, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.
• e-a4c5b7fb551c21c9 veronese_section (r=2, j=0, shift 0) → a(n) = 1 if n is a triangular number, otherwise 0.

Related sequences ?

Same fingerprint (75)

Equal up to content, sign, reversal and a small index shift.

• A023531 → (θ+1)² → D
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
• A131431: 3n + 1 preceded by n zeros.
• A136572 → θ+1
• basis-map coefficients of A026729, shifted → (θ+1)²
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
• basis-map coefficients of A297191, shifted → D
• basis-map coefficients of A026729, shifted → θ+1
• basis-map coefficients of A026729 → (θ+1)²
• A038338 → γ
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• A010054 → reverse → (θ+1)²
• basis-map coefficients of A297191, shifted → θ+1
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• basis-map coefficients of A112227 → D
• A130123 → reverse
• basis-map coefficients of A112227 → (θ+1)²
• A023531 → 2D+1+2n → Veronese 2 (j=1)
• A178820 → γ
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A130123 → (θ+1)²
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
• A178822 → γ
• A023531 → (θ+1)² → θ+1
• basis-map coefficients of A026729, shifted → reverse
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
• A010054 → reverse → D
• basis-map coefficients of A scaled Hermite triangle.
• basis-map coefficients of A021012 → reverse
• A010054 → reverse → D
• A028326 → γ
• basis-map coefficients of A030528, shifted → D
• A344565 → γ
• A156992 → γ
• A010054 → reverse → reverse
• A038221 → γ
• basis-map coefficients of A026729, shifted → D
• A010054 → reverse → reverse
• basis-map coefficients of A026729 → D
• basis-map coefficients of A021012 → θ+1
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• basis-map coefficients of A112227 → θ+1
• A010054 → reverse → D
• A023531 → (θ+1)² → reverse
• basis-map coefficients of A030528, shifted → reverse
• A130123 → D
• A010054 → reverse → θ+1
• A136572 → D
• A085881 → γ
• A136572 → (θ+1)²
• basis-map coefficients of A297191, shifted → (θ+1)²
• A023531 → θ+1
• A356546 → γ
• A023531 → (θ+1)² → (θ+1)²
• basis-map coefficients of A021012 → (θ+1)²
• A178821 → γ
• A133080 → Veronese 2
• A010054 → reverse → θ+1
• A135838 → γ
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A196347 → γ
• basis-map coefficients of A030528, shifted → θ+1
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• A023531 → 2D+1+2n → Veronese 2
• A023531 → D → (θ+1)²
• A007318 → D → γ
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
• A130123 → θ+1
• A038325 → γ
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• A010054 → reverse → θ+1
• A010054 → reverse → (θ+1)²
• basis-map coefficients of A021012 → D
• basis-map coefficients of A030528, shifted → (θ+1)²
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
4. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
5. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
6. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.