A038208 → D+1

OEIS matches ?

• A292044: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:    4,     2;
  2:   12,    16,      4;
  3:   32,    72,     48,      8;
  4:   80,   256,    288,    128,     16;
  5:  192,   800,   1280,    960,    320,     32;
  6:  448,  2304,   4800,   5120,   2880,    768,     64;
  7: 1024,  6272,  16128,  22400,  17920,   8064,   1792,   128;
  8: 2304, 16384,  50176,  86016,  89600,  57344,  21504,  4096,  256;
  9: 5120, 41472, 147456, 301056, 387072, 322560, 172032, 55296, 9216, 512;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (2*n + 4)*x + 2*x^2)*P[n - 1, x])/(n + 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] = ((2*n + 2 + (2*n + 4)*x + 2*x^2)*v[i - 1])/(n + 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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-8aab31d568b7b201
• 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-8aab31d568b7b201
• 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-8aab31d568b7b201
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-8aab31d568b7b201

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 derivative_polynomial, shift 0 e-8aab31d568b7b201

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4f4e9585c5795aa9 root: theorem check

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

    • parent_provers periodic_product e-0d998323de038265 root: theorem check

      Parent-row provers of polytool polytool ?

    • basis_map A007318 e-1de26a40255316f7

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-7963c2f0e7b86ba8 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-37a63c273567eea5

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)
    • preserver_image reversal, shift 0 e-cf1dcb38aee4f2af

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-4f4e9585c5795aa9 (expanded above)
        • ↵ parent_provers periodic_product e-0d998323de038265 (expanded above)
        • ↵ basis_map A007318 e-1de26a40255316f7 (expanded above)
        • basis_map A013609 e-e548dab430fb29bc

          Geometric basis map (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 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 0 e-07e3ff75f72fdf91

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-37a63c273567eea5 (expanded above)
    • ↵ basis_map A013609 e-e548dab430fb29bc (expanded above)
    • ↵ preserver_image affine, shift 0 e-07e3ff75f72fdf91 (expanded above)

Related sequences ?

Same fingerprint (27)

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

• A007318 → ×(1+2n+n²) → D+1
• A344565 → D+1
• A007318 → ×(1+2n+n²) → θ+1
• A178821 → θ+1
• A085880 → θ+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → θ+1
• A028326 → D+1
• A085880 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A007318 → D → θ+1
• A135838 → D+1
• A178820 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A178820 → D+1
• A178821 → D+1
• A135838 → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A028326 → θ+1
• basis-map coefficients of A113214, shifted → D+1
• A007318 → D → D+1
• A344565 → θ+1
• A178822 → D+1
• A038208 → θ+1
• A178822 → θ+1
• basis-map coefficients of A113214, shifted → θ+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → D+1

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