A038333 → x ↦ x+1 → D+1

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Triangle begins:
  0:             627,              364,               49;
  1:           14440,            13167,             3822,              343;
  2:          322373,           404320,           184338,            35672,             2401;
  3:         7037334,         11283055,          7075600,          2150610,           312130,            16807;
  4:       151042039,        295568028,        236944155,         99058400,         22581405,          2621892,          117649;
  5:      3199119908,       7401059911,       7241416686,       3870087865,       1213465400,        221297769,        21412118,         823543;
  6:     67040380425,     179150714848,     207229677508,     135173111472,      54181230110,      13590812480,      2065445844,      171296944,        5764801;
  7:   1392652169362,    4223543966775,    5643247517712,    4351823227668,    2128976505684,     682683499386,    142703531040,    18589012596,     1348963434,      40353607;
  8:  28719205102331,   97485651855340,  147824038837125,  131675775413280,   76156906484190,   29805671079576,   7964640826170,  1427035310400,   162653860215,   10491937820,   282475249;
  9: 588582360748896, 2211378792879487, 3753197596430590, 3794150330152875, 2534758676705640, 1172816359856526, 382506112187892, 87611049087870, 13735214862600, 1391594137395, 80787921214, 1977326743;

The row polynomials satisfy (7*n+26+7*x)*P(n,x) = (133*n+627+(49*n+364)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 627 + 364*x + 49*x^2.

P[0, x_] := 627 + 364*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((133*n + 627 + (49*n + 364)*x + 49*x^2)*P[n - 1, x])/(7*n + 26 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 627 + 364*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((133*n + 627 + (49*n + 364)*x + 49*x^2)*v[i - 1])/(7*n + 26 + 7*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)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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)*7^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (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.