basis-map coefficients of A127647 → basis A013610

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Triangle begins:
  0:     1;
  1:     2,      3;
  2:     8,     24,      18;
  3:    24,    108,     162,      81;
  4:    80,    480,    1080,    1080,      405;
  5:   256,   1920,    5760,    8640,     6480,     1944;
  6:   832,   7488,   28080,   56160,    63180,    37908,     9477;
  7:  2688,  28224,  127008,  317520,   476280,   428652,   214326,    45927;
  8:  8704, 104448,  548352, 1645056,  3084480,  3701376,  2776032,  1189728,  223074;
  9: 28160, 380160, 2280960, 7983360, 17962560, 26943840, 26943840, 17321040, 6495390, 1082565;

The row polynomials satisfy P(n,x) = (4+12*x+9*x^2)*P(n-2,x) + (2+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[(4 + 12*x + 9*x^2)*P[n - 2, x] + (2 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 12*x + 9*x^2)*v[i - 2] + (2 + 3*x)*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 (−∞, 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 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 ?

• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A038243 → basis A013611
• A038221 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A038338 → basis A013610 → reverse
• A094305 → basis A013610
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A013610 → D → basis A013609
• A038334 → reverse
• A028326 → basis A013610 → D
• A013609 → basis A013610 → x ↦ x+1
• A027465 → basis A013611 → basis A013610
• A085881 → basis A013609 → basis A013610
• A038325 → basis A013609 → basis A013610
• A178821 → basis A013609 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A038220 → D
• A178821 → basis A013611 → basis A007318
• A003506 → D → basis A013610
• A135838 → basis A013610
• A028326 → basis A013609 → basis A013610
• A038208 → basis A013610
• A038330 → basis A013611 → basis A013610
• A038208 → D → basis A013610
• A085880 → basis A013610 → reverse
• A178820 → basis A013610
• A038208 → basis A013610 → D
• A344565 → basis A013610
• A156992 → basis A013610 → D
• A156992 → basis A013611 → x ↦ x+1
• A085880 → basis A013609 → basis A013610
• A196347 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A136572 → basis A013609 → x ↦ x+1
• A134346 → basis A013610
• A178820 → basis A007318 → basis A013609
• A038243 → basis A013611 → D
• A131431 → x ↦ x+1 → basis A013610
• A356546 → x ↦ x+1 → basis A013609
• A003506 → x ↦ x+1 → basis A013609
• A156992 → basis A013610
• A178822 → basis A013609 → basis A013610
• basis-map coefficients of A113214, shifted → D → basis A013610
• A013610 → basis A013611 → basis A013609
• A178820 → basis A013609 → basis A013610
• A356546 → basis A013611 → basis A007318
• A085881 → basis A013610
• A120910 → x ↦ x/(x+1)
• A132200 → basis A013610
• A178822 → x ↦ x+1 → basis A013609
• A120910 → basis A013609
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A356546 → basis A013610
• A196347 → basis A013610 → D
• A136572 → basis A013610 → basis A013609
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A120910 → D → basis A013609
• A178822 → basis A013610 → D
• A038334 → D
• A178821 → basis A013610
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A085880 → basis A013611 → x ↦ x+1
• A196347 → basis A013611 → x ↦ x+1
• A085881 → basis A007318 → basis A013609
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A028326 → basis A007318 → basis A013609
• A134058 → basis A013610
• A135065 → basis A013610
• A028326 → D → basis A013610
• A356546 → basis A013609 → basis A013610
• A038221 → basis A013609 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A135089 → basis A013610
• A134059 → basis A013610
• A178822 → basis A013610
• A178822 → basis A013611 → basis A007318
• A028326 → basis A013611 → x ↦ x+1
• A120910 → basis A013609 → reverse
• A013620 → D
• A027466 → basis A013611 → x ↦ x/(x+1)
• A135065 → x ↦ x+1 → basis A013609
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A038338 → basis A013609 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A038291 → basis A013609 → basis A013611
• A038221 → basis A013611 → x ↦ x+1
• A124860 → basis A013610
• A038325 → basis A007318 → basis A013609
• A258758 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A156992 → basis A013609 → basis A013610
• A178820 → basis A013611 → basis A007318
• A085881 → basis A013611 → basis A007318
• A178821 → basis A013610 → D
• A028326 → basis A013610
• A094305 → x ↦ x+1 → basis A013609
• A085880 → basis A013610 → D
• A038330 → basis A013609 → basis A013609
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A120910 → basis A013609 → D
• A085880 → basis A013610
• A376827 → basis A013610
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A038330 → basis A013609 → x ↦ x/(x+1)
• A178820 → basis A013610 → D
• A038338 → basis A013610
• A120910 → basis A013610 → basis A013610
• A038221 → basis A013610 → D
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• A216973 → basis A013610
• A038328 → basis A013609 → basis A013610
• A038338 → basis A013610 → D
• A038325 → basis A013610
• A156992 → basis A007318 → basis A013609
• A085880 → D → basis A013610
• A178821 → basis A013610 → reverse

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.