A139548 → θ+1

Matches ?

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Submission text ?

Triangle begins:
  0:   1;
  1:   2,     8,      6;
  2:   4,    32,     72,      64,      20;
  3:   8,    96,    360,     640,     600,      288,       56;
  4:  16,   256,   1344,    3584,    5600,     5376,     3136,      1024,       144;
  5:  32,   640,   4320,   15360,   33600,    48384,    47040,     30720,     12960,      3200,       352;
  6:  64,  1536,  12672,   56320,  158400,   304128,   413952,    405504,    285120,    140800,     46464,      9216,       832;
  7: 128,  3584,  34944,  186368,  640640,  1537536,  2690688,   3514368,   3459456,   2562560,   1409408,    559104,    151424,    25088,     1920;
  8: 256,  8192,  92160,  573440, 2329600,  6709248, 14350336,  23429120,  29652480,  29286400,  22550528,  13418496,   6056960,  2007040,   460800,   65536,    4352;
  9: 512, 18432, 235008, 1671168, 7833600, 26320896, 66533376, 130351104, 201636864, 248934400, 246445056, 195526656, 123561984, 61415424, 23500800, 6684672, 1331712, 165888, 9728;

The row polynomials satisfy (1+(2*n-1)*x)*P(n,x) = (2+(4*n+6)*x+(8*n+6)*x^2+(4*n+2)*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[((2 + (4*n + 6)*x + (8*n + 6)*x^2 + (4*n + 2)*x^3)*P[n - 1, x])/(1 + (2*n - 1)*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] = ((2 + (4*n + 6)*x + (8*n + 6)*x^2 + (4*n + 2)*x^3)*v[i - 1])/(1 + (2*n - 1)*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 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)

Related ?

Same fingerprint ?

• A034870 → θ+1
• A034870 → D+1
• A034870 → basis A013609 → D+1
• A139548 → basis A013609 → D+1
• A139548 → D+1
• A139548 → basis A013609 → θ+1
• A034870 → basis A013609 → 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.