A376827 → basis A013610

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Submission text ?

Triangle begins:
  0:       1;
  1:       2,        3;
  2:      12,       36,        27;
  3:      56,      252,       378,       189;
  4:     304,     1824,      4104,      4104,       1539;
  5:    1632,    12240,     36720,     55080,      41310,      12393;
  6:    9024,    81216,    304560,    609120,     685260,     411156,     102789;
  7:   50304,   528192,   2376864,   5942160,    8913240,    8021916,    4010958,    859491;
  8:  283392,  3400704,  17853696,  53561088,  100427040,  120512448,   90384336,  38736144,   7263027;
  9: 1607168, 21696768, 130180608, 455632128, 1025172288, 1537758432, 1537758432, 988558992, 370709622, 61784937;

The row polynomials satisfy n*P(n,x) = (12*n-12+(36*n-36)*x+(27*n-27)*x^2)*P(n-2,x) + (4*n-2+(6*n-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[Together[((12*n - 12 + (36*n - 36)*x + (27*n - 27)*x^2)*P[n - 2, x] + (4*n - 2 + (6*n - 3)*x)*P[n - 1, x])/(n)]];
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] = ((12*n - 12 + (36*n - 36)*x + (27*n - 27)*x^2)*v[i - 2] + (4*n - 2 + (6*n - 3)*x)*v[i - 1])/(n)); 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 T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
    • 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 T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A134346 → basis A013610
• A135838 → basis A013610
• A028326 → basis A013609 → basis A013610
• A178822 → basis A013610 → D
• A135065 → basis A013610
• A356546 → basis A013610
• A178821 → basis A013609 → basis A013610
• A156992 → basis A013610 → D
• A085880 → basis A013610 → D
• A085880 → basis A013611 → x ↦ x+1
• A344565 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A085881 → basis A013609 → basis A013610
• A094305 → x ↦ x+1 → basis A013609
• A038208 → D → basis A013610
• A094305 → basis A013610
• A038221 → basis A013609 → basis A013610
• A038338 → basis A013610
• A135065 → x ↦ x+1 → basis A013609
• A085881 → basis A013611 → basis A007318
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A038334 → reverse
• A003506 → x ↦ x+1 → basis A013609
• A178820 → basis A013610 → D
• A120910 → D → basis A013609
• A028326 → basis A007318 → basis A013609
• A085880 → basis A013610 → reverse
• A038208 → basis A013610
• basis-map coefficients of A113214, shifted → D → basis A013610
• A038221 → basis A013611 → x ↦ x+1
• A085880 → D → basis A013610
• A038243 → basis A013611
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A038330 → basis A013611 → basis A013610
• A178821 → basis A013610 → D
• A038330 → basis A013609 → x ↦ x/(x+1)
• A178822 → basis A013611 → basis A007318
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A085881 → basis A007318 → basis A013609
• A003506 → D → basis A013610
• A178822 → basis A013609 → basis A013610
• A038325 → basis A013609 → basis A013610
• A013609 → basis A013610 → x ↦ x+1
• A156992 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• A038221 → basis A013610 → D
• A038221 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A132200 → basis A013610
• A156992 → basis A013609 → basis A013610
• A156992 → basis A013611 → x ↦ x+1
• A085880 → basis A013610
• A178822 → x ↦ x+1 → basis A013609
• A038328 → basis A013609 → basis A013610
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A038338 → basis A013610 → D
• A038208 → basis A013610 → D
• A134059 → basis A013610
• A178822 → basis A013610
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A258758 → basis A013610
• A038338 → basis A013610 → reverse
• A124860 → basis A013610
• A038243 → basis A013611 → D
• A027465 → basis A013611 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A120910 → basis A013610 → basis A013610
• A038291 → basis A013609 → basis A013611
• A085881 → basis A013610
• A038338 → basis A013609 → basis A013610
• A120910 → basis A013609 → D
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A178821 → basis A013610
• A028326 → basis A013611 → x ↦ x+1
• A196347 → basis A013610 → D
• A196347 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A085880 → basis A013609 → basis A013610
• A120910 → x ↦ x/(x+1)
• A135089 → basis A013610
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A356546 → basis A013609 → basis A013610
• A196347 → basis A013611 → x ↦ x+1
• A120910 → basis A013609
• A013620 → D
• A178820 → basis A013611 → basis A007318
• A038334 → D
• A027466 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A216973 → basis A013610
• A028326 → basis A013610 → D
• A038325 → basis A013610
• A028326 → D → basis A013610
• A178820 → basis A013610
• A136572 → basis A013610 → basis A013609
• A038330 → basis A013609 → basis A013609
• A013610 → basis A013611 → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A120910 → basis A013609 → reverse
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A038220 → D
• A156992 → basis A007318 → basis A013609
• A356546 → basis A013611 → basis A007318
• basis-map coefficients of A127647 → basis A013610
• A131431 → x ↦ x+1 → basis A013610
• A356546 → x ↦ x+1 → basis A013609
• A178820 → basis A013609 → basis A013610
• A134058 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A038325 → basis A007318 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A178821 → basis A013610 → reverse
• A178821 → basis A013611 → basis A007318
• A028326 → basis A013610
• A178820 → basis A007318 → basis A013609
• A136572 → basis A013609 → x ↦ x+1
• A013610 → D → basis A013609

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.