A038331 → x ↦ x+1 → D+1

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Triangle begins:
  0:             459,             220,              25;
  1:            9248,            6885,            1650,             125;
  2:          181781,          184960,           68850,           11000,             625;
  3:         3507882,         4544525,         2312000,          573750,           68750,           3125;
  4:        66733279,       105236460,        68167875,        23120000,         4303125,         412500,          15625;
  5:      1255153588,      2335664765,      1841638050,       795291875,       202300000,       30121875,        2406250,         78125;
  6:     23389304361,     50206143520,     46713295300,     24555174000,      7952918750,     1618400000,      200812500,      13750000,       390625;
  7:    432496961342,   1052518696245,   1129638229200,    700699429500,    276245707500,    71576268750,    12138000000,    1290937500,     77343750,     1953125;
  8:   7945387725299,  21624848067100,  26312967406125,  18827303820000,   8758742868750,  2762457075000,   596468906250,   86700000000,   8068359375,   429687500,    9765625;
  9: 145151560832328, 436996324891445, 594683321845250, 482404402445625, 258875427525000, 96346171556250, 25322523187500, 4686541406250, 596062500000, 49306640625, 2363281250, 48828125;

The row polynomials satisfy (5*n+22+5*x)*P(n,x) = (85*n+459+(25*n+220)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 459 + 220*x + 25*x^2.

P[0, x_] := 459 + 220*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((85*n + 459 + (25*n + 220)*x + 25*x^2)*P[n - 1, x])/(5*n + 22 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 459 + 220*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((85*n + 459 + (25*n + 220)*x + 25*x^2)*v[i - 1])/(5*n + 22 + 5*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)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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)*5^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (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.