A038331 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           264,            170,             25;
  1:          3888,           3960,           1275,            125;
  2:         55296,          77760,          39600,           8500,            625;
  3:        767232,        1382400,         972000,         330000,          53125,           3125;
  4:      10450944,       23016960,       20736000,        9720000,        2475000,         318750,         15625;
  5:     140341248,      365783040,      402796800,      241920000,       85050000,       17325000,       1859375,         78125;
  6:    1863254016,     5613649920,     7315660800,     5370624000,     2419200000,      680400000,     115500000,      10625000,       390625;
  7:   24508956672,    83846430720,   126307123200,   109734912000,    60419520000,    21772800000,    5103000000,     742500000,     59765625,     1953125;
  8:  319906381824,  1225447833600,  2096160768000,  2105118720000,  1371686400000,   604195200000,  181440000000,   36450000000,   4640625000,   332031250,    9765625;
  9: 4148463403008, 17594851000320, 33699815424000, 38429614080000, 28945382400000, 15088550400000, 5538456000000, 1425600000000, 250593750000, 28359375000, 1826171875, 48828125;

The row polynomials satisfy (5*n+17+5*x)*P(n,x) = (60*n+264+(25*n+170)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 264 + 170*x + 25*x^2.

P[0, x_] := 264 + 170*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 264 + (25*n + 170)*x + 25*x^2)*P[n - 1, x])/(5*n + 17 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 264 + 170*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 264 + (25*n + 170)*x + 25*x^2)*v[i - 1])/(5*n + 17 + 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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • 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

Related ?

Same fingerprint ?

• A038271 → basis A007318 → 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.