basis-map coefficients of A297191, shifted → D → basis A013611

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:  0;
  1:  2,    8;
  2:  4,   48,   192,    256;
  3:  6,  120,   960,   3840,     7680,      6144;
  4:  8,  224,  2688,  17920,    71680,    172032,    229376,     131072;
  5: 10,  360,  5760,  53760,   322560,   1290240,   3440640,    5898240,     5898240,      2621440;
  6: 12,  528, 10560, 126720,  1013760,   5677056,  22708224,   64880640,   129761280,    173015040,    138412032,     50331648;
  7: 14,  728, 17472, 256256,  2562560,  18450432,  98402304,  393609216,  1180827648,   2624061440,   4198498304,   4580179968,    3053453312,     939524096;
  8: 16,  960, 26880, 465920,  5591040,  49201152, 328007680, 1686896640,  6747586560,  20992491520,  50381979648,  91603599360,  122138132480,  112742891520,   64424509440,   17179869184;
  9: 18, 1224, 39168, 783360, 10967040, 114057216, 912457728, 5735448576, 28677242880, 114708971520, 367068708864, 934356713472, 1868713426944, 2874943733760, 3285649981440, 2628519985152, 1314259992576, 309237645312;

The row polynomials satisfy (n-1)*P(n,x) = (n+8*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 8*x.

P[0, x_] := 0;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 8*n*x + 16*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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 8*n*x + 16*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delanno…?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A297191, shifted → basis A013611 → 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.