A038319 → basis A013610 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:             736,              930,              225;
  1:           15616,            33120,            20925,              3375;
  2:          311296,           936960,           993600,            418500,             50625;
  3:         5963776,         23347200,         35136000,          24840000,           7846875,            759375;
  4:       111149056,        536739840,       1050624000,        1054080000,         558900000,         141243750,          11390625;
  5:      2030043136,      11670650880,      28178841600,       36771840000,       27669600000,       11736900000,        2471765625,         170859375;
  6:     36507222016,     243605176320,     700239052800,     1127153664000,     1103155200000,      664070400000,      234738000000,       42373125000,       2562890625;
  7:    648540061696,    4928474972160,   16443349401600,    31510757376000,    38041436160000,    29785190400000,    14941584000000,     4527090000000,     715046484375,      38443359375;
  8:  11407433138176,   97281009254400,  369635622912000,   822167470080000,  1181653401600000,  1141243084800000,   744629760000000,   320176800000000,   84882937500000,   11917441406250,    576650390625;
  9: 199011604627456, 1882226467799040, 8025683263488000, 20329959260160000, 33914408140800000, 38994562252800000, 31384184832000000, 17551987200000000, 6603646500000000, 1556187187500000, 196637783203125, 8649755859375;

The row polynomials satisfy (15*n+31+15*x)*P(n,x) = (240*n+736+(225*n+930)*x+225*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 736 + 930*x + 225*x^2.

P[0, x_] := 736 + 930*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((240*n + 736 + (225*n + 930)*x + 225*x^2)*P[n - 1, x])/(15*n + 31 + 15*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 736 + 930*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((240*n + 736 + (225*n + 930)*x + 225*x^2)*v[i - 1])/(15*n + 31 + 15*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 whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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
        • Geometric basis map (elementary) polytool ? step ›

          • from basis_map_A007318 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 Triangle of coefficients in expansion of (1+5x)^n.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.