A139548 → D

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Submission text ?

Triangle begins:
  0:    0;
  1:    4,      4;
  2:   16,     48,      48,      16;
  3:   48,    240,     480,     480,      240,       48;
  4:  128,    896,    2688,    4480,     4480,     2688,       896,       128;
  5:  320,   2880,   11520,   26880,    40320,    40320,     26880,     11520,      2880,       320;
  6:  768,   8448,   42240,  126720,   253440,   354816,    354816,    253440,    126720,     42240,      8448,       768;
  7: 1792,  23296,  139776,  512512,  1281280,  2306304,   3075072,   3075072,   2306304,   1281280,    512512,    139776,    23296,     1792;
  8: 4096,  61440,  430080, 1863680,  5591040, 12300288,  20500480,  26357760,  26357760,  20500480,  12300288,   5591040,  1863680,   430080,   61440,    4096;
  9: 9216, 156672, 1253376, 6266880, 21934080, 57028608, 114057216, 179232768, 224040960, 224040960, 179232768, 114057216, 57028608, 21934080, 6266880, 1253376, 156672, 9216;

The row polynomials satisfy (n-1)*P(n,x) = (2*n+4*n*x+2*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 4*x.

P[0, x_] := 0;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 4*n*x + 2*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 4*n*x + 2*n*x^2)*v[i - 1])/(n - 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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 → D
• A034870 → D → basis A013609
• A034870 → basis A013609 → D
• A139548 → basis A013609 → D

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.