A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1

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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:     0;
  1:     0;
  2:     4,     4,      1;
  3:    24,    36,     18,      3;
  4:    96,   192,    144,     48,      6;
  5:   320,   800,    800,    400,    100,    10;
  6:   960,  2880,   3600,   2400,    900,   180,    15;
  7:  2688,  9408,  14112,  11760,   5880,  1764,   294,   21;
  8:  7168, 28672,  50176,  50176,  31360, 12544,  3136,  448,  28;
  9: 18432, 82944, 165888, 193536, 145152, 72576, 24192, 5184, 648, 36;

The row polynomials satisfy P(n,x) = (8+12*x+6*x^2+x^3)*P(n-3,x) - (12+12*x+3*x^2)*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4 + 4*x + x^2.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 4 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 12*x + 6*x^2 + x^3)*P[n - 3, x] - (12 + 12*x + 3*x^2)*P[n - 2, x] + (6 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 12*x + 6*x^2 + x^3)*v[i - 3] - (12 + 12*x + 3*x^2)*v[i - 2] + (6 + 3*x)*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, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-7ca9df991456824b
• 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-7ca9df991456824b
• 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-7ca9df991456824b

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 affine, shift 0 e-7ca9df991456824b

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(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]
    • preserver_image reversal, shift 0 e-a72beb3d57d8ea71

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(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]
        • preserver_image euler_polynomial, shift 0 e-ccfad80d3bf960c1

          Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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 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 euler_polynomial, shift 0 e-ccfad80d3bf960c1 (expanded above)

Related sequences ?

Same fingerprint (67)

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

• A038207 → D
• A028326 → basis A007318
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038330 → basis A013609
• A028326 → basis A013611
• A120909 → D
• A085880 → x ↦ x+1
• A196347 → basis A013611
• A038338 → basis A007318
• A038325 → basis A013611
• A085880 → x ↦ x/(x+1)
• A013609 → D
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A156992 → basis A013611
• A178822 → x ↦ x+1
• A178820 → basis A007318
• A038338 → basis A013611
• A120909 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A356546 → basis A007318
• A038221 → basis A013611
• basis-map coefficients of A021012 → basis A013609
• A159854 → x ↦ x/(x+1)
• A085881 → basis A007318
• basis-map coefficients of A113214, shifted → basis A013611
• basis-map coefficients of A026729 → basis A013609
• A038325 → basis A007318
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• basis-map coefficients of A030528, shifted → basis A013609
• A196347 → basis A007318
• A178822 → basis A013611
• A135838 → x ↦ x+1
• A344565 → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A130123 → basis A013609
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A136572 → basis A013609
• basis-map coefficients of A112227 → basis A013609
• A344565 → basis A007318
• A120910 → D
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A085881 → basis A013611
• A356546 → basis A013611
• A038243 → basis A013610
• A178822 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013609
• A013609 → basis A013610
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A344565 → basis A013611
• A038208 → basis A013611
• A135838 → x ↦ x/(x+1)
• A178820 → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A178821 → basis A007318
• A007318 → D → x ↦ x/(x+1)
• A135838 → basis A013611
• A178821 → x ↦ x/(x+1)
• A085880 → basis A013611
• A027466 → basis A013611
• A038208 → basis A007318
• A028326 → x ↦ x/(x+1)
• A178821 → basis A013611
• A038208 → x ↦ x/(x+1)
• A156992 → basis A007318
• A178820 → basis A013611

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