A143491 → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:          40,           60,           16;
  1:         248,          572,          384,           64;
  2:        1728,         5176,         5360,         2176,          256;
  3:       13536,        48616,        65840,        41280,        11520,         1024;
  4:      118368,       487136,       796384,       656320,       282880,        58368,         4096;
  5:     1145952,      5250528,      9893856,      9894976,      5626880,      1792000,       286720,       16384;
  6:    12185280,     60947136,    128512704,    148859648,    103160064,     43466752,     10723328,     1376256,       65536;
  7:   141295680,    760643136,   1758580800,   2289139712,   1844650752,    950412288,    310861824,    61440000,     6488064,     262144;
  8:  1775381760,  10178249472,  25424153856,  36422570240,  33101864960,  19897371648,   7979962368,  2094366720,   340131840,   30146560,   1048576;
  9: 24037966080, 145562294016, 388511735040, 603327321344, 605402772480, 411677133824, 193159938048, 62302175232, 13451919360, 1831075840, 138412032, 4194304;

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(8*n+24)*x+16*x^2)*P(n-2,x) + (2*n+7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 40 + 60*x + 16*x^2, P(1,x) = 248 + 572*x + 384*x^2 + 64*x^3.

P[0, x_] := 40 + 60*x + 16*x^2;
P[1, x_] := 248 + 572*x + 384*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (8*n + 24)*x + 16*x^2)*P[n - 2, x] + (2*n + 7 + 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] = 40 + 60*x + 16*x^2; v[2] = 248 + 572*x + 384*x^2 + 64*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (8*n + 24)*x + 16*x^2)*v[i - 2] + (2*n + 7 + 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 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 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 → 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.