A038331 → basis A013610 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:              1024,                960,                225;
  1:             32768,              46080,              21600,               3375;
  2:           1048576,            1966080,            1382400,             432000,              50625;
  3:          33554432,           78643200,           73728000,           34560000,            8100000,             759375;
  4:        1073741824,         3019898880,         3538944000,         2211840000,          777600000,          145800000,           11390625;
  5:       34359738368,       112742891520,       158544691200,       123863040000,        58060800000,        16329600000,         2551500000,         170859375;
  6:     1099511627776,      4123168604160,      6764573491200,      6341787648000,      3715891200000,      1393459200000,       326592000000,       43740000000,        2562890625;
  7:    35184372088832,    148434069749760,    278313880780800,    304405807104000,    214035333120000,    100329062400000,     31352832000000,     6298560000000,      738112500000,      38443359375;
  8:  1125899906842624,   5277655813324800,  11132555231232000,  13915694039040000,  11415217766400000,   6421059993600000,   2508226560000000,   671846400000000,   118098000000000,   12301875000000,    576650390625;
  9: 36028797018963968, 185773484629032960, 435406604599296000, 612290537717760000, 574022379110400000, 376702186291200000, 176579149824000000, 59122483200000000, 13856832000000000, 2165130000000000, 202980937500000, 8649755859375;

The row polynomials satisfy P(n,x) = (32+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 960*x + 225*x^2.

P[0, x_] := 1024 + 960*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 15*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1024 + 960*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 15*x)*v[i - 1]); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • ↑ Geometric basis map (elementary) (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.