A038338 → basis A013610 → basis A008277

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Submission text ?

Triangle begins:
  0:                 3600,                   5616,                    1296;
  1:               216000,                 668736,                  373248,                    46656;
  2:             12960000,               71974656,                73343232,                 21275136,                  1679616;
  3:            777600000,             7376126976,             12272394240,               6214579200,               1108546560,                 60466176;
  4:          46656000000,           736101789696,           1885496610816,            1485855498240,             449868349440,              54419558400,               2176782336;
  5:        2799360000000,         72345387810816,         275385217056768,          317501601693696,          145263941222400,           29255954595840,            2559896027136,             78364164096;
  6:      167961600000000,       7045934189838336,       38955282612682752,        63254136898584576,        41063901670342656,        12250930987008000,         1759745668939776,         116605876174848,          2821109907456;
  7:    10077696000000000,     682456299824480256,     5395750935708303360,     12029085173018787840,     10654184869098946560,      4418523897014255616,       926723320159666176,       99731877448384512,       5179557790089216,       101559956668416;
  8:   604661760000000000,   65878601839150104576,   736807550307177332736,   2215133342752655278080,   2606500779524861460480,   1443996390570983424000,    414842496656581066752,    64478385289643950080,    5392833699092889600,    225463103803883520,    3656158440062976;
  9: 36279705600000000000, 6346113599918410039296, 99630226306756811685888, 398667473393504470695936, 611470959362167327948800, 440393161799931034337280, 166480399137771777687552, 35051586090005830828032, 4197927997711507783680, 280719845028035297280, 9652258281766256640, 131621703842267136;

The row polynomials satisfy P(n,x) = (60+36*x)*P(n-1,x) + 36*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3600 + 5616*x + 1296*x^2.

P[0, x_] := 3600 + 5616*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 36*x)*P[n - 1, x] + 36*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3600 + 5616*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 36*x)*v[i - 1] + 36*x*deriv(v[i - 1], 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

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

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • Geometric basis map (elementary) polytool ? step ›

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

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

Related ?

Same fingerprint ?

• A178820 → basis A013610 → basis A008277
• A196347 → basis A013610 → basis A008277
• A225466 → D+1
• A178822 → basis A013610 → basis A008277
• A028326 → basis A013610 → basis A008277
• A038208 → basis A013610 → basis A008277
• A013620 → basis A008277
• A156992 → basis A013610 → basis A008277
• A038221 → basis A013610 → basis A008277
• A085880 → basis A013610 → basis A008277
• A178821 → basis A013610 → basis A008277

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.