A038336 → θ+1

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Submission text ?

Triangle begins:
  0:          144,            480,            300;
  1:         1728,           8640,          10800,            4000;
  2:        20736,         138240,         259200,          192000,           50000;
  3:       248832,        2073600,        5184000,         5760000,         3000000,          600000;
  4:      2985984,       29859840,       93312000,       138240000,       108000000,        43200000,         7000000;
  5:     35831808,      418037760,     1567641600,      2903040000,      3024000000,      1814400000,       588000000,        80000000;
  6:    429981696,     5733089280,    25082265600,     55738368000,     72576000000,     58060800000,     28224000000,      7680000000,       900000000;
  7:   5159780352,    77396705280,   386983526400,   1003290624000,   1567641600000,   1567641600000,   1016064000000,    414720000000,     97200000000,    10000000000;
  8:  61917364224,  1031956070400,  5804752896000,  17199267840000,  31352832000000,  37623398400000,  30481920000000,  16588800000000,   5832000000000,  1200000000000,   110000000000;
  9: 743008370688, 13621820129280, 85136375808000, 283787919360000, 591224832000000, 827714764800000, 804722688000000, 547430400000000, 256608000000000, 79200000000000, 14520000000000, 1200000000000;

The row polynomials satisfy (6+(5*n+10)*x)*P(n,x) = (72+(60*n+240)*x+(50*n+150)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 480*x + 300*x^2.

P[0, x_] := 144 + 480*x + 300*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72 + (60*n + 240)*x + (50*n + 150)*x^2)*P[n - 1, x])/(6 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 480*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (60*n + 240)*x + (50*n + 150)*x^2)*v[i - 1])/(6 + (5*n + 10)*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.?
    • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.?
        • 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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038259 → θ+1
• A013612 → x ↦ x+1 → θ+1
• A038271 → basis A013609 → θ+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.