A395972 → reverse

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Submission text ?

Triangle begins:
   1:           8,           16,             4;
   2:          48,          144,            72,             8;
   3:         384,         1536,          1152,           256,            16;
   4:        3840,        19200,         19200,          6400,           800,           32;
   5:       46080,       276480,        345600,        153600,         28800,         2304,          64;
   6:      645120,      4515840,       6773760,       3763200,        940800,       112896,        6272,        128;
   7:    10321920,     82575360,     144506880,      96337920,      30105600,      4816896,      401408,      16384,       256;
   8:   185794560,   1672151040,    3344302080,    2601123840,     975421440,    195084288,    21676032,    1327104,     41472,      512;
   9:  3715891200,  37158912000,   83607552000,   74317824000,   32514048000,   7803371520,  1083801600,   88473600,   4147200,   102400,   1024;
  10: 81749606400, 899245670400, 2248114176000, 2248114176000, 1124057088000, 314735984640, 52455997440, 5352652800, 334540800, 12390400, 247808, 2048;

The row polynomials satisfy P(n,x) = (2*n+2+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 8 + 16*x + 4*x^2.

P[1, x_] := 8 + 16*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2*n + 2 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 8 + 16*x + 4*x^2; for(n = 2, N, my(i = n); v[i] = (2*n + 2 + 2*x)*v[i - 1] + 2*x*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, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).?
    • 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
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A130534 → basis A008277 → θ+1
• A271703 → D
• A132393 → basis A008277
• A395972 → D
• A144084 → D
• Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
• A105278 → θ+1
• A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
• A271703 → 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.