A143491 → basis A013610 → reverse

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Submission text ?

Triangle begins:
  0:      9,      21,       12;
  1:     27,     108,      141,        60;
  2:     81,     486,     1071,      1026,        360;
  3:    243,    2025,     6615,     10575,       8262,       2520;
  4:    729,    8019,    36045,     84645,     109386,      73656,       20160;
  5:   2187,   30618,   180306,    578340,    1089963,    1205442,      723384,      181440;
  6:   6561,  113724,   847098,   3538080,    9053289,   14515956,    14224572,     7778160,     1814400;
  7:  19683,  413343,  3792258,  19932318,   66078747,  143134047,   202349232,   179804772,    91002960,    19958400;
  8:  59049, 1476225, 16336890, 105304050,  437424057, 1222347105,  2324656260,  2967605100,  2430666144,  1151910720,   239500800;
  9: 177147, 5196312, 68201595, 528291720, 2681224821, 9353554056, 22864481145, 39123346680, 45870864732, 35054392032, 15693341760, 3113510400;

The row polynomials satisfy P(n,x) = (3+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 21*x + 12*x^2.

P[0, x_] := 9 + 21*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + (n + 4)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 21*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + (n + 4)*x)*v[i - 1]); 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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A143491 → basis A013610
• A136124 → basis A013610
• A136124 → basis A013610 → reverse

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.