A038338 → basis A013609 → basis A008277

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Submission text ?

Triangle begins:
  0:                2304,                  2880,                    576;
  1:              110592,                262656,                 124416,                  13824;
  2:             5308416,              21565440,               18247680,                4644864,                  331776;
  3:           254803968,            1680113664,             2269347840,              995328000,               159252480,                7962624;
  4:         12230590464,          127083479040,           258180120576,           173903708160,             46820229120,             5159780352,              191102976;
  5:        587068342272,         9443544662016,         27835295072256,         27064767873024,          10915801989120,          1990528598016,           160526499840,            4586471424;
  6:      28179280429056,       694024855879680,       2898833398824960,       3915819226497024,        2221429915385856,        596388052205568,         78593774321664,         4843313823744,         110075314176;
  7:    1352605460594688,     50646092353634304,     294944602828308480,     539470308751441920,      413865569251491840,     153507510739795968,      29403317922693120,      2932406369648640,      142657607172096,       2641807540224;
  8:   64925062108545024,   3678981180515942400,   29530188088004837376,   71815107518052433920,    72543929382249431040,   35722035466321526784,    9329617298912182272,   1339079405988741120,   104615578592870400,    4121219762749440,     63403380965376;
  9: 3116402981210161152, 266444846487752933376, 2923193604780847005696, 9326537416278408167424, 12169888411477176483840, 7742356263516002844672, 2648615372582855639040, 513151966867460456448, 57245644606018682880, 3598775903594741760, 117169448024014848, 1521681143169024;

The row polynomials satisfy P(n,x) = (48+24*x)*P(n-1,x) + 24*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2304 + 2880*x + 576*x^2.

P[0, x_] := 2304 + 2880*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(48 + 24*x)*P[n - 1, x] + 24*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] = 2304 + 2880*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (48 + 24*x)*v[i - 1] + 24*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • ↑ Geometric basis map (elementary) (above)
        • 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)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A178822 → basis A013609 → basis A008277
• A038208 → D → basis A008277
• basis-map coefficients of A021012 → basis A008277 → D+1
• A028326 → D → basis A008277
• A178821 → basis A008277
• A028326 → basis A013609 → basis A008277
• A135838 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A038208 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A085880 → basis A008277
• A134059 → basis A008277
• A356546 → basis A013609 → basis A008277
• A134346 → basis A008277
• A028326 → basis A008277
• A156992 → basis A008277 → reverse
• A178820 → basis A008277
• A003506 → D → basis A008277
• A038325 → basis A008277
• A178821 → basis A013609 → basis A008277
• A038221 → basis A013609 → basis A008277
• basis-map coefficients of A127647 → basis A008277
• A178820 → basis A013609 → basis A008277
• A132200 → basis A008277
• A131431 → x ↦ x+1 → basis A008277
• A038325 → basis A013609 → basis A008277
• A156992 → basis A008277
• A156992 → basis A013609 → basis A008277
• A085880 → basis A013609 → basis A008277
• A003506 → basis A008277
• A094305 → basis A008277
• A196347 → basis A013609 → basis A008277
• A376827 → basis A008277
• A027465 → basis A013611 → basis A008277
• A038221 → basis A008277
• A135065 → basis A008277
• basis-map coefficients of A112227 → x ↦ x+1 → basis A008277
• A126351 → reverse
• A143494 → reverse
• A356546 → basis A008277
• A120910 → basis A013610 → basis A008277
• A344565 → basis A008277
• A196347 → basis A008277
• A134058 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A085881 → basis A013609 → basis A008277
• A038330 → basis A013611 → basis A008277
• A085880 → D → basis A008277
• A178822 → basis A008277
• A135089 → basis A008277
• basis-map coefficients of A026729, shifted → basis A007318 → basis A008277
• A085881 → basis A008277
• A038338 → basis A008277
• A258758 → basis A008277
• basis-map coefficients of A113214, shifted → D → basis A008277
• basis-map coefficients of A113214, shifted → basis A013609 → basis A008277
• A124860 → 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.