A038305 → basis A013611 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:              625,               600,               144;
  1:            15625,             22500,             10800,              1728;
  2:           390625,            750000,            540000,            172800,             20736;
  3:          9765625,          23437500,          22500000,          10800000,           2592000,            248832;
  4:        244140625,         703125000,         843750000,         540000000,         194400000,          37324800,           2985984;
  5:       6103515625,       20507812500,       29531250000,       23625000000,       11340000000,        3265920000,         522547200,         35831808;
  6:     152587890625,      585937500000,      984375000000,      945000000000,      567000000000,      217728000000,       52254720000,       7166361600,        429981696;
  7:    3814697265625,    16479492187500,    31640625000000,    35437500000000,    25515000000000,    12247200000000,     3919104000000,     806215680000,      96745881600,      5159780352;
  8:   95367431640625,   457763671875000,   988769531250000,  1265625000000000,  1063125000000000,   612360000000000,   244944000000000,   67184640000000,   12093235200000,   1289945088000,    61917364224;
  9: 2384185791015625, 12588500976562500, 30212402343750000, 43505859375000000, 41765625000000000, 28066500000000000, 13471920000000000, 4618944000000000, 1108546560000000, 177367449600000, 17027275161600, 743008370688;

The row polynomials satisfy P(n,x) = (25+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 600*x + 144*x^2.

P[0, x_] := 625 + 600*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(25 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 625 + 600*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 12*x)*v[i - 1]); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(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 basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • ↑ Geometric basis map (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.