A143491 → basis A013611 → basis A013609

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Submission text ?

Triangle begins:
  0:           56,           120,             64;
  1:          504,          1528,           1536,            512;
  2:         5040,         19312,          27584,          17408,           4096;
  3:        55440,        252752,         457920,         412160,         184320,          32768;
  4:       665280,       3476544,        7517056,        8609280,        5509120,        1867776,         262144;
  5:      8648640,      50517312,      125534080,      172057088,      140492800,       68354048,       18350080,        2097152;
  6:    121080960,     776431488,     2161615616,     3413071872,     3343355904,     2080899072,      803733504,      176160768,      16777216;
  7:   1816214400,   12615120000,    38635686144,    68489003008,    77454913536,    57960333312,    28703195136,     9072279552,    1660944384,     134217728;
  8:  29059430400,  216371635200,   719091938304,  1404909537280,  1787190640640,  1547004641280,   922933788672,   374782033920,   99153346560,   15435038720,   1073741824;
  9: 494010316800, 3910793241600, 13955536032768, 29636197640192, 41621517189120, 40596604026880, 28065911537664, 13754764886016, 4683863162880, 1055622430720, 141733920768, 8589934592;

The row polynomials satisfy P(n,x) = (n+8+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 56 + 120*x + 64*x^2.

P[0, x_] := 56 + 120*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 8 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 56 + 120*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 8 + 8*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 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 ?

• A136124 → basis A013611 → basis A013609

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.