A060821: Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.

OEIS matches ?

• A060821: triangle, 66 terms agree, 11 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A062267: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:    0,     2;
  2:   -2,     0,      4;
  3:    0,   -12,      0,      8;
  4:   12,     0,    -48,      0,    16;
  5:    0,   120,      0,   -160,     0,    32;
  6: -120,     0,    720,      0,  -480,     0,    64;
  7:    0, -1680,      0,   3360,     0, -1344,     0,   128;
  8: 1680,     0, -13440,      0, 13440,     0, -3584,     0, 256;
  9:    0, 30240,      0, -80640,     0, 48384,     0, -9216,   0, 512;

The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - 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[2*x*P[n - 1, x] - D[P[n - 1, x], 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*x*v[i - 1] - deriv(v[i - 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)

Triangle begins:
  0:    1;
  1:    0,     2;
  2:   -2,     0,      4;
  3:    0,   -12,      0,      8;
  4:   12,     0,    -48,      0,    16;
  5:    0,   120,      0,   -160,     0,    32;
  6: -120,     0,    720,      0,  -480,     0,    64;
  7:    0, -1680,      0,   3360,     0, -1344,     0,   128;
  8: 1680,     0, -13440,      0, 13440,     0, -3584,     0, 256;
  9:    0, 30240,      0, -80640,     0, 48384,     0, -9216,   0, 512;

The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - 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[2*x*P[n - 1, x] - D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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*x*v[i - 1] - deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5f70bba9f74a761e

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

• basis_map A060821 e-5f70bba9f74a761e

  Hermite-type basis map [MW08, Thm 2] 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)

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. [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