A038328 → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:            420,             352,              64;
  1:           7448,           10080,            4224,             512;
  2:         126224,          238336,          161280,           45056,            4096;
  3:        2074464,         5048960,         4766720,         2150400,          450560,           32768;
  4:       33345088,        99574272,       121175040,        76267520,        25804800,         4325376,          262144;
  5:      527067520,      1867324928,      2788079616,      2261934080,      1067745280,       289013760,        40370176,        2097152;
  6:     8222253312,     33732321280,     59754397696,     59479031808,     36190945280,     13667139584,      3082813440,      369098752,       16777216;
  7:   126917858816,    592002238464,   1214363566080,   1434105544704,   1070622572544,    521149612032,    164005675008,    31708938240,     3321888768,     134217728;
  8:  1942138397696,  10153428705280,  23680089538560,  32383028428800,  28682110894080,  17129961160704,   6948661493760,  1874350571520,   317089382400,   29527900160,   1073741824;
  9: 29503974807552, 170908178997248, 446750863032320, 694615959797760, 712426625433600, 504805151735808, 251239430356992, 87354601635840, 20617856286720, 3100429516800, 259845521408, 8589934592;

The row polynomials satisfy (4*n+11+4*x)*P(n,x) = (56*n+210+(32*n+176)*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 420 + 352*x + 64*x^2.

P[0, x_] := 420 + 352*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((56*n + 210 + (32*n + 176)*x + 32*x^2)*P[n - 1, x])/(4*n + 11 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 420 + 352*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((56*n + 210 + (32*n + 176)*x + 32*x^2)*v[i - 1])/(4*n + 11 + 4*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 basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) 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 → 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.