A007318 → basis A049020 → x ↦ x/(x+1)

OEIS matches ?

No OEIS matches.

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:      2,       3;
  2:      5,      15,      11;
  3:     15,      67,      98,       47;
  4:     52,     307,     670,      641,      227;
  5:    203,    1486,    4299,     6146,     4344,     1215;
  6:    877,    7648,   27501,    52204,    55191,    30823,     7107;
  7:   4140,   41847,  179612,   424411,   596399,   498541,   229606,    44959;
  8:  21147,  242857, 1210098,  3417413,  5983741,  6653261,  4588620,  1795165,   305091;
  9: 115975, 1490395, 8448371, 27728366, 58078061, 80519503, 73904431, 43312372, 14710196, 2206399;

The row polynomials satisfy (1+x)*P(n,x) = -(n-3+(4*n-9)*x+(5*n-8)*x^2+(2*n-2)*x^3)*P(n-1,x) + (1+5*x+9*x^2+7*x^3+2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 3 + (4*n - 9)*x + (5*n - 8)*x^2 + (2*n - 2)*x^3)*P[n - 1, x] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*D[P[n - 1, x], x])/(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] = (-(n - 3 + (4*n - 9)*x + (5*n - 8)*x^2 + (2*n - 2)*x^3)*v[i - 1] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + x)); 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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-426e25db4b42e73d
• 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-426e25db4b42e73d
• 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-426e25db4b42e73d

Proof chains

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

• preserver_image mobius, shift 0 e-426e25db4b42e73d

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: A008277 * A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A049020 e-3c112fda27f20cb0

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

                  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 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)
                • 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)
            • 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 affine, shift 1 e-62a7be97cd0c2a0d

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-995393ce4e49a8d6 root: theorem check

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

        • basis_map A008277 e-c52960bc5e97caa0

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

          • 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-d527982dc7c0aa3b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-8e580efc86a38a12 root: theorem check

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

            • preserver_image reversal, shift 0 e-3256729839b26765

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-995393ce4e49a8d6 (expanded above)
                • ↵ basis_map A008277 e-c52960bc5e97caa0 (expanded above)
    • preserver_image derivative_polynomial, shift 0 e-9acfc4c7c0e80a75

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

      • premise (sequence): Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • basis_map A049020 e-2ff0affdc4d5d50d

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

          • 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)
        • preserver_image affine, shift 0 e-bcbfbcbb753a93dd

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • basis_map A008277 e-857a86cbb6c7881e

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

              • premise (sequence): basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
                • preserver_image affine, shift 0 e-0ba23af98afc4615

                  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)
                • preserver_image affine, shift 0 e-8c1f314a0c6d10ae

                  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)
            • preserver_image reversal, shift 1 e-ad13a5d17ca116d8

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • theorem_check liu_wang e-2ab1dfe057e84c9d root: theorem check

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

Related sequences ?

No candidate equalities or related sequences.

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. [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
4. [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
5. [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
6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.