A066325: Hermite he

OEIS matches ?

• A066325: triangle, 78 terms agree, 12 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 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,    1;
  2:  -1,    0,    1;
  3:   0,   -3,    0,     1;
  4:   3,    0,   -6,     0,   1;
  5:   0,   15,    0,   -10,   0,   1;
  6: -15,    0,   45,     0, -15,   0,   1;
  7:   0, -105,    0,   105,   0, -21,   0,   1;
  8: 105,    0, -420,     0, 210,   0, -28,   0, 1;
  9:   0,  945,    0, -1260,   0, 378,   0, -36, 0, 1;

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

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (n - 1)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] - (n - 1)*v[i - 2]); 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,    1;
  2:  -1,    0,    1;
  3:   0,   -3,    0,     1;
  4:   3,    0,   -6,     0,   1;
  5:   0,   15,    0,   -10,   0,   1;
  6: -15,    0,   45,     0, -15,   0,   1;
  7:   0, -105,    0,   105,   0, -21,   0,   1;
  8: 105,    0, -420,     0, 210,   0, -28,   0, 1;
  9:   0,  945,    0, -1260,   0, 378,   0, -36, 0, 1;

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

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (n - 1)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] - (n - 1)*v[i - 2]); 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-5cce3c36fe7dc594, e-74d857940dd4e27e, e-b31a65521e9db5ca
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-5cce3c36fe7dc594
• 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-74d857940dd4e27e

Proof chains

4 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 A066325 e-b31a65521e9db5ca

  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)
• definition_equality e-12f19e4cfb6ce810 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image euler_polynomial, shift 0 e-74d857940dd4e27e

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

  • premise (sequence): hermite_he
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • theorem_check three_term_recurrence e-5cce3c36fe7dc594 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • ↵ basis_map A066325 e-b31a65521e9db5ca (expanded above)
• ↵ theorem_check three_term_recurrence e-5cce3c36fe7dc594 (expanded above)

Used in 5 proof steps

• e-74d857940dd4e27e preserver_image (euler_polynomial, shift 0) → hermite_he
• e-9c6e37fd70f39446 preserver_image (affine, shift -1) → Coefficient array of Hermite_H(n, (x-1)/sqrt(2))/(sqrt(2))^n.
• e-d30779985c323b83 preserver_image (affine, shift 0) → shift_x_plus_one of hermite_he
• e-f8533e45cc68d6b0 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of hermite_he
• e-f9b0b1f7e619a5c6 preserver_image (derivative_polynomial, shift 0) → derivative of hermite_he

Related sequences ?

Same fingerprint (1)

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

• A066325 → D

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
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.
8. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023