A139548 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
  0:         1;
  1:         8,         48,           54;
  2:        64,        768,         2592,          3456,           1620;
  3:       512,       9216,        51840,        138240,         194400,         139968,           40824;
  4:      4096,      98304,       774144,       3096576,        7257600,       10450944,         9144576,         4478976,           944784;
  5:     32768,     983040,      9953280,      53084160,      174182400,      376233984,       548674560,       537477120,        340122240,        125971200,         20785248;
  6:    262144,    9437184,    116785152,     778567680,     3284582400,     9459597312,     19313344512,     28378791936,      29930757120,      22170931200,      10974610944,       3265173504,        442158912;
  7:   2097152,   88080384,   1288175616,   10305404928,    53137244160,   191294078976,    502146957312,    983798120448,    1452639412224,    1614043791360,    1331586127872,     792348770304,     321891687936,      79996750848,       9183300480;
  8:  16777216,  805306368,  13589544960,  126835752960,   772905369600,  3338951196672,  10712468422656,  26234616545280,   49804779847680,   73784859033600,   85221512183808,   76065481949184,   51502670069760,   25598960271360,    8815968460800,   1880739938304,    187339329792;
  9: 134217728, 7247757312, 138613358592, 1478542491648, 10396001894400, 52395849547776, 198667596201984, 583839466389504, 1354690011856896, 2508685207142400, 3725397532606464, 4433530947895296, 4202617877692416, 3133312737214464, 1798457566003200, 767341894828032, 229303339665408, 42845606719488, 3768826516992;

The row polynomials satisfy (2+(6*n-3)*x)*P(n,x) = (16+(48*n+72)*x+(144*n+108)*x^2+(108*n+54)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16 + (48*n + 72)*x + (144*n + 108)*x^2 + (108*n + 54)*x^3)*P[n - 1, x])/(2 + (6*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16 + (48*n + 72)*x + (144*n + 108)*x^2 + (108*n + 54)*x^3)*v[i - 1])/(2 + (6*n - 3)*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 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

Proof chains ? JSON

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

  • from basis_map_A013610 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • 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
            • 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
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A034870 → basis A013610 → θ+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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.