A038329 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:            315,            108,              9;
  1:           5400,           2835,            486,             27;
  2:          91125,          64800,          17010,           1944,             81;
  3:        1518750,        1366875,         486000,          85050,           7290,           243;
  4:       25059375,       27337500,       12301875,        2916000,         382725,         26244,          729;
  5:      410062500,      526246875,      287043750,       86113125,       15309000,       1607445,        91854,        2187;
  6:     6663515625,     9841500000,     6314962500,     2296350000,      516678750,      73483200,      6429780,      314928,       6561;
  7:   107641406250,   179914921875,   132860250000,    56834662500,    15500362500,    2790065250,    330674400,    24800580,    1062882,     19683;
  8:  1729951171875,  3229242187500,  2698723828125,  1328602500000,   426259968750,   93002175000,  13950326250,  1417176000,   93002175,   3542940,    59049;
  9: 27679218750000, 57088388671875, 53282496093750, 29685962109375, 10960970625000, 2813315793750, 511511962500, 65765823750, 5845851000, 341007975, 11691702, 177147;

The row polynomials satisfy (n+6+x)*P(n,x) = (15*n+105+(3*n+36)*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 315 + 108*x + 9*x^2.

P[0, x_] := 315 + 108*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n + 105 + (3*n + 36)*x + 3*x^2)*P[n - 1, x])/(n + 6 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 315 + 108*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n + 105 + (3*n + 36)*x + 3*x^2)*v[i - 1])/(n + 6 + 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)*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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038243 → D+1
• A038291 → basis A013609 → D+1
• A147716 → basis A013610 → 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.