A038330 → basis A013611 → basis A008277

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Submission text ?

Triangle begins:
  0:              1024,                1280,                  256;
  1:             32768,               77824,                36864,                  4096;
  2:           1048576,             4259840,              3604480,                917504,                 65536;
  3:          33554432,           221249536,            298844160,             131072000,              20971520,              1048576;
  4:        1073741824,         11156848640,          22666018816,           15267266560,            4110417920,            452984832,             16777216;
  5:       34359738368,        552708603904,        1629134782464,         1584037625856,          638876385280,         116500987904,           9395240960,           268435456;
  6:     1099511627776,      27079768801280,      113107963740160,       152789166587904,        86676735000576,       23270132809728,        3066606649344,        188978561024,         4294967296;
  7:    35184372088832,    1317421088505856,     7672185980190720,     14032860746874880,     10765593225461760,     3993082634698752,      764847776071680,      76278619176960,      3710851743744,       68719476736;
  8:  1125899906842624,   63799162201702400,   512098640148299776,   1245383835433041920,   1258022721594327040,   619474746693648384,   161789837492355072,   23221685578629120,   1814194185830400,    71468255805440,    1099511627776;
  9: 36028797018963968, 3080374184191197184, 33795099564718424064, 107824285077016150016, 140696322639984066560, 89509535175197851648, 30620695146119823360, 5932560123200274432, 661818038990929920, 41605519995043840, 1354598325420032, 17592186044416;

The row polynomials satisfy P(n,x) = (32+16*x)*P(n-1,x) + 16*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1024 + 1280*x + 256*x^2.

P[0, x_] := 1024 + 1280*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 16*x)*P[n - 1, x] + 16*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] = 1024 + 1280*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 16*x)*v[i - 1] + 16*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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013611 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
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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

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.