A038329 → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:            585,             648,              144;
  1:          11475,           21060,            11664,             1728;
  2:         212625,          550800,           505440,           186624,            20736;
  3:        3796875,        12757500,         16524000,         10108800,          2799360,           248832;
  4:       66065625,       273375000,        459270000,        396576000,        181958400,         40310784,          2985984;
  5:     1127671875,      5549512500,      11481750000,      12859560000,       8328096000,       3056901120,        564350976,         35831808;
  6:    18965390625,    108256500000,     266376600000,     367416000000,     308629440000,     159899443200,      48910417920,       7739670528,       429981696;
  7:   315235546875,   2048262187500,    5845851000000,    9589557600000,    9920232000000,    6666395904000,    2878189977600,     754617876480,    104485552128,      5159780352;
  8:  5189853515625,  37828265625000,  122895731250000,  233834040000000,  287686728000000,  238085568000000,  133327918080000,   49340399616000,  11319268147200,   1393140695040,    61917364224;
  9: 84767607421875, 685060664062500, 2496665531250000, 5407412175000000, 7716523320000000, 7594929619200000, 5237882496000000, 2514183598080000, 814116593664000, 166015932825600, 18389457174528, 743008370688;

The row polynomials satisfy (4*n+9+4*x)*P(n,x) = (60*n+195+(48*n+216)*x+48*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 585 + 648*x + 144*x^2.

P[0, x_] := 585 + 648*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 195 + (48*n + 216)*x + 48*x^2)*P[n - 1, x])/(4*n + 9 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 585 + 648*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 195 + (48*n + 216)*x + 48*x^2)*v[i - 1])/(4*n + 9 + 4*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • 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 ?

• A038334 → basis A013609 → D+1
• A013611 → basis A007318 → 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.