A038284 → basis A013610 → basis A013609

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Submission text ?

Triangle begins:
  0:              1024,               2304,                1296;
  1:             32768,             110592,              124416,               46656;
  2:           1048576,            4718592,             7962624,             5971968,              1679616;
  3:          33554432,          188743680,           424673280,           477757440,            268738560,             60466176;
  4:        1073741824,         7247757312,         20384317440,         30576476160,          25798901760,          11609505792,           2176782336;
  5:       34359738368,       270582939648,        913217421312,       1712282664960,        1926317998080,        1300264648704,         487599243264,          78364164096;
  6:     1099511627776,      9895604649984,      38963943309312,      87668872445952,      123284351877120,      110955916689408,       62412703137792,       20061226008576,        2821109907456;
  7:    35184372088832,    356241767399424,    1603087953297408,    4208105877405696,     7101178668122112,     7988826001637376,     5991619501228032,     2888816545234944,      812479653347328,     101559956668416;
  8:  1125899906842624,  12666373951979520,   64123518131896320,  192370554395688960,   378729528966512640,   511284864104792064,   479329560098242560,   308140431491727360,   129996744535572480,   32499186133893120,    3656158440062976;
  9: 36028797018963968, 445856363109679104, 2507942042491944960, 8464304393410314240, 19044684885173207040, 29995378694147801088, 33744801030916276224, 27116357971272007680, 15252951358840504320, 5719856759565189120, 1286967770902167552, 131621703842267136;

The row polynomials satisfy P(n,x) = (32+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 2304*x + 1296*x^2.

P[0, x_] := 1024 + 2304*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 36*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 + 2304*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 36*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 (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from 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)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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 basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
            • ↑ Geometric basis map (elementary) (above)
            • 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)*2^j.?
                • Geometric basis map (elementary) polytool ? step ›

                  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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) (above)

Related ?

Same fingerprint ?

• A038335 → basis A013610 → basis A013609
• A038322 → basis A007318 → basis A013610
• A038334 → basis A013609 → basis A013609
• A038329 → basis A013611 → basis A013609
• A013620 → basis A007318 → basis A013610
• A027467 → basis A013609 → x ↦ x/(x+1)
• A013611 → basis A007318 → basis A013609
• A038279 → x ↦ x/(x+1)
• A038329 → basis A013610 → basis A013610

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.