basis-map coefficients of A127647 → basis A008277

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Triangle begins:
  0:     1;
  1:     2,       1;
  2:     8,      10,       2;
  3:    24,      57,      27,        3;
  4:    80,     325,     275,       70,       5;
  5:   256,    1688,    2280,     1000,     160,       8;
  6:   832,    8645,   17563,    11830,    3185,     351,     13;
  7:  2688,   43239,  127449,   123921,   49980,    9114,    735,    21;
  8:  8704,  214370,  895390,  1209516,  686154,  184212,  24276,  1496,   34;
  9: 28160, 1054405, 6140475, 11231275, 8616300, 3195885, 612150, 61050, 2970, 55;

The row polynomials satisfy P(n,x) = (4+5*x+x^2)*P(n-2,x) + (5*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(4 + 5*x + x^2)*P[n - 2, x] + (5*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}] + (2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 5*x + x^2)*v[i - 2] + (5*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), x) + (2 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).?
    • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A135065 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A132200 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A126351 → reverse
• A156992 → basis A013609 → basis A008277
• A028326 → basis A008277
• A156992 → basis A008277
• basis-map coefficients of A026729, shifted → basis A007318 → basis A008277
• A038338 → basis A008277
• A178822 → basis A013609 → basis A008277
• A028326 → basis A013609 → basis A008277
• A038325 → basis A013609 → basis A008277
• A003506 → D → basis A008277
• A038221 → basis A013609 → basis A008277
• A085881 → basis A013609 → basis A008277
• A178821 → basis A013609 → basis A008277
• A120910 → basis A013610 → basis A008277
• A028326 → D → basis A008277
• A356546 → basis A008277
• A131431 → x ↦ x+1 → basis A008277
• A134346 → basis A008277
• A135838 → basis A008277
• basis-map coefficients of A113214, shifted → D → basis A008277
• A196347 → basis A008277
• A196347 → basis A013609 → basis A008277
• A038330 → basis A013611 → basis A008277
• A135089 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• basis-map coefficients of A021012 → basis A008277 → D+1
• A038221 → basis A008277
• A094305 → basis A008277
• A038338 → basis A013609 → basis A008277
• A258758 → basis A008277
• A178820 → basis A008277
• A085880 → D → basis A008277
• A143494 → reverse
• A178822 → basis A008277
• A156992 → basis A008277 → reverse
• A038325 → basis A008277
• A124860 → basis A008277
• A003506 → basis A008277
• A376827 → basis A008277
• A038208 → D → basis A008277
• A085880 → basis A013609 → basis A008277
• A356546 → basis A013609 → basis A008277
• A134058 → basis A008277
• A178820 → basis A013609 → basis A008277
• A085880 → basis A008277
• A027465 → basis A013611 → basis A008277
• basis-map coefficients of A113214, shifted → basis A013609 → basis A008277
• A038208 → basis A008277
• A178821 → basis A008277
• basis-map coefficients of A112227 → x ↦ x+1 → basis A008277
• A085881 → basis A008277
• A134059 → basis A008277
• A344565 → basis A008277

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.