basis-map coefficients of A297191, shifted → D → basis A013609

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:  0;
  1:  2,   4;
  2:  4,  24,   48,    32;
  3:  6,  60,  240,   480,    480,     192;
  4:  8, 112,  672,  2240,   4480,    5376,     3584,     1024;
  5: 10, 180, 1440,  6720,  20160,   40320,    53760,    46080,     23040,      5120;
  6: 12, 264, 2640, 15840,  63360,  177408,   354816,   506880,    506880,    337920,    135168,     24576;
  7: 14, 364, 4368, 32032, 160160,  576576,  1537536,  3075072,   4612608,   5125120,   4100096,   2236416,    745472,    114688;
  8: 16, 480, 6720, 58240, 349440, 1537536,  5125120, 13178880,  26357760,  41000960,  49201152,  44728320,  29818880,  13762560,   3932160,   524288;
  9: 18, 612, 9792, 97920, 685440, 3564288, 14257152, 44808192, 112020480, 224040960, 358465536, 456228864, 456228864, 350945280, 200540160, 80216064, 20054016, 2359296;

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

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

• A034870 → basis A013611 → D
• A034870 → D → basis A007318
• A139548 → x ↦ x+1 → D
• A034870 → D → basis A013611
• 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.