A038338 → x ↦ x+1 → basis A008277

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Submission text ?

Triangle begins:
  0:               1296,                1008,                  144;
  1:              46656,               63936,                20736,                 1728;
  2:            1679616,             3628800,              2011392,               373248,                20736;
  3:           60466176,           194337792,            164229120,             51010560,              6220800,               248832;
  4:         2176782336,         10053808128,          12185800704,           5643509760,           1134673920,             98537472,             2985984;
  5:        78364164096,        508704178176,         851793739776,         552562311168,         163034726400,          23075684352,          1504935936,           35831808;
  6:      2821109907456,      25358170521600,       57212074524672,       50006011281408,       20325664751616,        4171682414592,        439441293312,        22359048192,         429981696;
  7:    101559956668416,    1251045503926272,     3737022517739520,     4286977706557440,     2307427974512640,      644389488820224,      97519848652800,      7956381302784,      325066162176,       5159780352;
  8:   3656158440062976,   61268903668482048,   239233897111486464,   353506665085009920,   245267682337751040,    89550526620893184,   18264817520345088,   2125003940167680,   138385309040640,    4643802316800,    61917364224;
  9: 131621703842267136, 2984781277367894016, 15089260670710972416, 28323286651458551808, 24844565297391206400, 11540062743658758144, 3047206611647987712, 474178283064262656, 43766908075376640, 2329331242106880, 65384736620544, 743008370688;

The row polynomials satisfy P(n,x) = (36+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1296 + 1008*x + 144*x^2.

P[0, x_] := 1296 + 1008*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 12*x)*P[n - 1, x] + 12*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] = 1296 + 1008*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 12*x)*v[i - 1] + 12*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_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) [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 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 ?

• A038330 → basis A013609 → basis A008277
• A038207 → basis A008277
• A178822 → x ↦ x+1 → basis A008277
• A120910 → basis A008277
• A356546 → x ↦ x+1 → basis A008277
• basis-map coefficients of A113214, shifted → x ↦ x+1 → basis A008277
• A028326 → basis A007318 → basis A008277
• A038221 → x ↦ x+1 → basis A008277
• A094305 → x ↦ x+1 → basis A008277
• A178821 → basis A007318 → basis A008277
• A038208 → basis A007318 → basis A008277
• A085880 → x ↦ x+1 → basis A008277
• A178820 → basis A007318 → basis A008277
• A038325 → basis A007318 → basis A008277
• A143495 → reverse
• A003506 → x ↦ x+1 → basis A008277
• A135065 → x ↦ x+1 → basis A008277
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
• A156992 → basis A007318 → basis A008277
• A038207 → basis A008277 → reverse
• A027466 → basis A013611 → basis A008277
• A196347 → x ↦ x+1 → basis A008277
• A120910 → D → basis A008277
• A085881 → basis A007318 → 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.