A013612 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:         96,         110,           25;
  1:        756,        1440,          825,          125;
  2:       5616,       15120,        14400,         5500,          625;
  3:      40176,      140400,       189000,       120000,        34375,         3125;
  4:     279936,     1205280,      2106000,      1890000,       900000,       206250,        15625;
  5:    1912896,     9797760,     21092400,     24570000,     16537500,      6300000,      1203125,        78125;
  6:   12877056,    76515840,    195955200,    281232000,    245700000,    132300000,     42000000,      6875000,      390625;
  7:   85660416,   579467520,   1721606400,   2939328000,   3163860000,   2211300000,    992250000,    270000000,    38671875,     1953125;
  8:  564350976,  4283020800,  14486688000,  28693440000,  36741600000,  31638600000,  18427500000,   7087500000,  1687500000,   214843750,    9765625;
  9: 3688436736, 31039303680, 117783072000, 265589280000, 394534800000, 404157600000, 290020500000, 144787500000, 48726562500, 10312500000, 1181640625, 48828125;

The row polynomials satisfy (5*n+11+5*x)*P(n,x) = (30*n+96+(25*n+110)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 96 + 110*x + 25*x^2.

P[0, x_] := 96 + 110*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 96 + (25*n + 110)*x + 25*x^2)*P[n - 1, x])/(5*n + 11 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 96 + 110*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((30*n + 96 + (25*n + 110)*x + 25*x^2)*v[i - 1])/(5*n + 11 + 5*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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038259 → D+1
• A038271 → basis A013609 → D+1
• A038336 → D+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.