A038328 → x ↦ x+1 → (θ+1)²

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Submission text ?

Triangle begins:
  0:           196,            224,             36;
  1:          2744,           4704,           1512,            128;
  2:         38416,          87808,          42336,           7168,            400;
  3:        537824,        1536640,         987840,         250880,          28000,          1152;
  4:       7529536,       25815552,       20744640,        7024640,        1176000,         96768,         3136;
  5:     105413504,      421654016,      406594944,      172103680,       38416000,       4741632,       307328,         8192;
  6:    1475789056,     6746464256,     7589772288,     3855122432,     1075648000,     177020928,     17210368,       917504,      20736;
  7:   20661046784,   106256812032,   136615901184,    80957571072,    27106329600,    5576159232,    722835456,     57802752,    2612736,     51200;
  8:  289254654976,  1652883742720,  2390778270720,  1619151421440,   632481024000,  156132458496,  25299240960,   2697461760,  182891520,   7168000,   123904;
  9: 4049565169664, 25454409637888, 40908872632320, 31168664862720, 13914582528000, 4007399768064, 779216621568, 103852277760, 9388431360, 551936000, 19081216, 294912;

The row polynomials satisfy (49+(21*n+35)*x+(n^2+4*n+4)*x^2)*P(n,x) = (686+(294*n+882)*x+(14*n^2+126*n+238)*x^2+(2*n^2+12*n+18)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 224*x + 36*x^2.

P[0, x_] := 196 + 224*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((686 + (294*n + 882)*x + (14*n^2 + 126*n + 238)*x^2 + (2*n^2 + 12*n + 18)*x^3)*P[n - 1, x])/(49 + (21*n + 35)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 224*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (294*n + 882)*x + (14*n^2 + 126*n + 238)*x^2 + (2*n^2 + 12*n + 18)*x^3)*v[i - 1])/(49 + (21*n + 35)*x + (n^2 + 4*n + 4)*x^2)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A007318 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A027466 → (θ+1)²

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.