A139548 → x ↦ x+1 → basis A007318

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Submission text ?

Triangle begins:
  0:            1;
  1:           18,            12,             2;
  2:          324,           432,           216,            48,             4;
  3:         5832,         11664,          9720,          4320,          1080,           144,             8;
  4:       104976,        279936,        326592,        217728,         90720,         24192,          4032,           384,            16;
  5:      1889568,       6298560,       9447840,       8398080,       4898880,       1959552,        544320,        103680,         12960,          960,           32;
  6:     34012224,     136048896,     249422976,     277136640,     207852480,     110854656,      43110144,      12317184,       2566080,       380160,        38016,        2304,         64;
  7:    612220032,    2857026816,    6190224768,    8253633024,    7565830272,    5043886848,    2521943424,     960740352,     280215936,     62270208,     10378368,     1257984,     104832,       5376,       128;
  8:  11019960576,   58773123072,  146932807680,  228562145280,  247608990720,  198087192576,  121053284352,   57644421120,   21616657920,   6404935680,   1494484992,   271724544,   37739520,    3870720,    276480,    12288,    256;
  9: 198359290368, 1190155742208, 3372107936256, 5994858553344, 7493573191680, 6994001645568, 5051223410688, 2886413377536, 1322939464704, 489977579520, 146993273856, 35634733056, 6928975872, 1065996288, 126904320, 11280384, 705024, 27648, 512;

The row polynomials satisfy P(n,x) = (18+12*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(18 + 12*x + 2*x^2)*P[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] = (18 + 12*x + 2*x^2)*v[i - 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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) 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)
• 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) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A297191, shifted → θ+1 → basis A013610
• basis-map coefficients of A297191, shifted → (θ+1)² → basis A013610
• A034870 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A297191, shifted → basis A013610
• basis-map coefficients of A297191, shifted → basis A013610 → reverse

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.