A139548 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:          0;
  1:          8,           4;
  2:        128,         192,          96,           16;
  3:       1536,        3840,        3840,         1920,          480,           48;
  4:      16384,       57344,       86016,        71680,        35840,        10752,         1792,          128;
  5:     163840,      737280,     1474560,      1720320,      1290240,       645120,       215040,        46080,         5760,         320;
  6:    1572864,     8650752,    21626880,     32440320,     32440320,     22708224,     11354112,      4055040,      1013760,      168960,       16896,        768;
  7:   14680064,    95420416,   286261248,    524812288,    656015360,    590413824,    393609216,    196804608,     73801728,    20500480,     4100096,     559104,      46592,      1792;
  8:  134217728,  1006632960,  3523215360,   7633633280,  11450449920,  12595494912,  10496245760,   6747586560,   3373793280,  1312030720,   393609216,   89456640,   14909440,   1720320,   122880,    4096;
  9: 1207959552, 10267656192, 41070624768, 102676561920, 179683983360, 233589178368, 233589178368, 183534354432, 114708971520, 57354485760, 22941794304, 7299661824, 1824915456, 350945280, 50135040, 5013504, 313344, 9216;

The row polynomials satisfy (n-1)*P(n,x) = (8*n+8*n*x+2*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8 + 4*x.

P[0, x_] := 0;
P[1, x_] := 8 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 8*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] = 8 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 8*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 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) 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 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 ?

• basis-map coefficients of A297191, shifted → D → basis A013609
• A034870 → D → basis A013611
• A034870 → basis A013611 → D
• A034870 → D → basis A007318
• basis-map coefficients of A297191, shifted → 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.