A038226 → basis A013610 → reverse

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Submission text ?

Triangle begins:
  0:              576,              528,               121;
  1:            13824,            19008,              8712,              1331;
  2:           331776,           608256,            418176,            127776,             14641;
  3:          7962624,         18247680,          16727040,           7666560,           1756920,            161051;
  4:        191102976,        525533184,         602173440,         367994880,         126498240,          23191344,          1771561;
  5:       4586471424,      14714929152,       20233027584,       15455784960,        7083901440,        1948072896,        297622248,         19487171;
  6:     110075314176,     403609485312,      647456882688,      593502142464,      340027269120,      124676665344,      28571735808,       3741536832,       214358881;
  7:    2641807540224,   10897456103424,    19978669522944,    21366077128704,    14689178025984,     6732539928576,    2057164978176,     404085977856,     46301518296,     2357947691;
  8:   63403380965376,  290598829424640,   599360085688320,   732551215841280,   587567121039360,   323161916571648,  123429898690560,   32326878228480,   5556182195520,   565907445840,   25937424601;
  9: 1521681143169024, 7671809096810496, 17581229180190720, 24174190122762240, 22159674279198720, 14219124329152512, 6517098650861568, 2133573963079680, 488944033205760, 74699782850880, 6847480094664, 285311670611;

The row polynomials satisfy P(n,x) = (24+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 528*x + 121*x^2.

P[0, x_] := 576 + 528*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 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] = 576 + 528*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 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_A013610 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 A013610
• A013622 → x ↦ x/(x+1) → basis A013610
• A038314 → basis A013611
• A038226 → basis A013610 → D

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.