A038328 → θ+1

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Submission text ?

Triangle begins:
  0:          144,            96,            12;
  1:         1728,          1728,           432,            32;
  2:        20736,         27648,         10368,          1536,           80;
  3:       248832,        414720,        207360,         46080,         4800,          192;
  4:      2985984,       5971968,       3732480,       1105920,       172800,        13824,         448;
  5:     35831808,      83607552,      62705664,      23224320,      4838400,       580608,       37632,       1024;
  6:    429981696,    1146617856,    1003290624,     445906944,    116121600,     18579456,     1806336,      98304,      2304;
  7:   5159780352,   15479341056,   15479341056,    8026324992,   2508226560,    501645312,    65028096,    5308416,    248832,     5120;
  8:  61917364224,  206391214080,  232190115840,  137594142720,  50164531200,  12039487488,  1950842880,  212336640,  14929920,   614400,   11264;
  9: 743008370688, 2724364025856, 3405455032320, 2270303354880, 945959731200, 264868724736, 51502252032, 7007109120, 656916480, 40550400, 1486848, 24576;

The row polynomials satisfy (6+(n+2)*x)*P(n,x) = (72+(12*n+48)*x+(2*n+6)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 96*x + 12*x^2.

P[0, x_] := 144 + 96*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72 + (12*n + 48)*x + (2*n + 6)*x^2)*P[n - 1, x])/(6 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 96*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (12*n + 48)*x + (2*n + 6)*x^2)*v[i - 1])/(6 + (n + 2)*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • 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
    • 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

Related ?

Same fingerprint ?

• A038255 → θ+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.