A038226 → basis A013611 → reverse

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Submission text ?

Triangle begins:
  0:              1024,                704,                121;
  1:             32768,              33792,              11616,               1331;
  2:           1048576,            1441792,             743424,             170368,              14641;
  3:          33554432,           57671680,           39649280,           13629440,            2342560,            161051;
  4:        1073741824,         2214592512,         1903165440,          872284160,          224885760,          30921792,           1771561;
  5:       34359738368,        82678120448,        85261811712,        48847912960,        16791470080,        3463240704,         396829664,         19487171;
  6:     1099511627776,      3023656976384,      3637837299712,      2501013143552,      1074654085120,      295529873408,       50794196992,       4988715776,        214358881;
  7:    35184372088832,    108851651149824,    149671020331008,    120048630890496,     61900075302912,    21278150885376,     4876242911232,     718375071744,      61735357728,      2357947691;
  8:  1125899906842624,   3870280929771520,   5986840813240320,   5487937412136960,   3301337349488640,  1361801656664064,   390099432898560,   76626674319360,    9877657236480,    754543261120,   25937424601;
  9: 36028797018963968, 136233888727957504, 234151996251176960, 241469246134026240, 166010106717143040, 79892363857625088, 27463000076058624, 6743147340103680, 1158978449080320, 132799613957120, 9129973459552, 285311670611;

The row polynomials satisfy P(n,x) = (32+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 704*x + 121*x^2.

P[0, x_] := 1024 + 704*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 11*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 + 704*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 11*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 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)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? 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
            • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038226 → basis A013611 → D
• A038226 → basis A013611
• A013622 → x ↦ x/(x+1) → basis A013611

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.