A038328 → D+1

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Submission text ?

Triangle begins:
  0:           192,            56,             4;
  1:          2592,          1152,           168,             8;
  2:         34560,         20736,          4608,           448,           16;
  3:        456192,        345600,        103680,         15360,         1120,          32;
  4:       5971968,       5474304,       2073600,        414720,        46080,        2688,          64;
  5:      77635584,      83607552,      38320128,       9676800,      1451520,      129024,        6272,        128;
  6:    1003290624,    1242169344,     668860416,     204374016,     38707200,     4644864,      344064,      14336,       256;
  7:   12899450880,   18059231232,   11179524096,    4013162496,    919683072,   139345920,    13934592,     884736,     32256,     512;
  8:  165112971264,  257989017600,  180592312320,   74530160640,  20065812480,  3678732288,   464486400,   39813120,   2211840,   71680,   1024;
  9: 2105190383616, 3632485367808, 2837879193600, 1324343623680, 409915883520, 88289574912, 13488685056, 1459814400, 109486080, 5406720, 157696, 2048;

The row polynomials satisfy (n+7+x)*P(n,x) = (12*n+96+(2*n+28)*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 192 + 56*x + 4*x^2.

P[0, x_] := 192 + 56*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 96 + (2*n + 28)*x + 2*x^2)*P[n - 1, x])/(n + 7 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 192 + 56*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 96 + (2*n + 28)*x + 2*x^2)*v[i - 1])/(n + 7 + 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 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 → 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.