A038338 → x ↦ x+1 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:             1152,               864,                144;
  1:            34560,             41472,              15552,               1728;
  2:           995328,           1658880,             995328,             248832,              20736;
  3:         27869184,          59719680,           49766400,           19906560,            3732480,            248832;
  4:        764411904,        2006581248,         2149908480,         1194393600,          358318080,          53747712,           2985984;
  5:      20639121408,       64210599936,        84276412416,        60197437440,        25082265600,        6019743744,         752467968,          35831808;
  6:     550376570880,     1981355655168,      3082108796928,      2696845197312,      1444738498560,      481579499520,       96315899904,       10319560704,        429981696;
  7:   14529941471232,    59440669655040,    106993205379072,    110955916689408,     72814820327424,    31206351568896,     8668430991360,     1486016741376,     139314069504,      5159780352;
  8:  380420285792256,  1743592976547840,   3566440179302400,   4279728215162880,   3328677500682240,  1747555687858176,   624127031377920,   148601674137600,   22290251120640,   1857520926720,    61917364224;
  9: 9890927430598656, 50215477724577792, 115077136452157440, 156923367889305600, 141231031100375040, 87877086018011136, 38446225132879872, 11769252591697920, 2451927623270400, 326923683102720, 24519276232704, 743008370688;

The row polynomials satisfy (n+3+x)*P(n,x) = (24*n+96+(12*n+72)*x+12*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1152 + 864*x + 144*x^2.

P[0, x_] := 1152 + 864*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 96 + (12*n + 72)*x + 12*x^2)*P[n - 1, x])/(n + 3 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1152 + 864*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 96 + (12*n + 72)*x + 12*x^2)*v[i - 1])/(n + 3 + 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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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
        • 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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A113214, shifted → x ↦ x+1 → D+1
• A178822 → x ↦ x+1 → D+1
• A094305 → x ↦ x+1 → D+1
• A085881 → basis A007318 → D+1
• A027466 → basis A013611 → D+1
• A003506 → x ↦ x+1 → D+1
• A093375 → D → basis A007318
• A120910 → D+1
• A038325 → basis A007318 → D+1
• A135065 → x ↦ x+1 → D+1
• A156992 → basis A007318 → D+1
• A196347 → x ↦ x+1 → D+1
• A178821 → basis A007318 → D+1
• A178820 → basis A007318 → D+1
• A103406 → x ↦ x/(x+1)
• A120910 → D → D+1
• A038330 → basis A013609 → D+1
• A093375 → x ↦ x+1 → reverse
• A085880 → x ↦ x+1 → D+1
• A038208 → basis A007318 → D+1
• A356546 → x ↦ x+1 → D+1
• A028326 → basis A007318 → D+1
• A038221 → x ↦ x+1 → D+1
• A038207 → D+1
• A093375 → x ↦ x+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.