A013612 → basis A013610 → D+1

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Submission text ?

Triangle begins:
  0:         216,          630,           225;
  1:        1836,         9720,         14175,           3375;
  2:       14256,       110160,        291600,         283500,          50625;
  3:      104976,      1069200,       4131000,        7290000,        5315625,          759375;
  4:      746496,      9447840,      48114000,      123930000,      164025000,        95681250,        11390625;
  5:     5178816,     78382080,     496011600,     1683990000,     3253162500,      3444525000,      1674421875,       170859375;
  6:    35271936,    621457920,    4702924800,    19840464000,    50519700000,     78075900000,     68890500000,     28704375000,      2562890625;
  7:   236825856,   4761711360,   41948409600,   211631616000,   669615660000,   1364031900000,   1756707750000,   1328602500000,    484386328125,     38443359375;
  8:  1572120576,  35523878400,  357128352000,  2097420480000,  7936185600000,  20088469800000,  34100797500000,  37643737500000,  24911296875000,   8073105468750,    576650390625;
  9: 10339716096, 259399895040, 2930719968000, 19642059360000, 86518594800000, 261894124800000, 552432919500000, 803804512500000, 776402085937500, 456707109375000, 133206240234375, 8649755859375;

The row polynomials satisfy (5*n+7+5*x)*P(n,x) = (30*n+72+(75*n+210)*x+75*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 216 + 630*x + 225*x^2.

P[0, x_] := 216 + 630*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 72 + (75*n + 210)*x + 75*x^2)*P[n - 1, x])/(5*n + 7 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 216 + 630*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((30*n + 72 + (75*n + 210)*x + 75*x^2)*v[i - 1])/(5*n + 7 + 5*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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • 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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
            • 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
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

• A013622 → basis A013611 → D+1

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.