A133214 → basis A013609 → basis A013609

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Submission text ?

Triangle begins:
  0:           61,           128,            64;
  1:          595,          1800,          1728,            512;
  2:         6145,         24320,         34560,          20480,           4096;
  3:        65527,        320840,        605440,         542720,         225280,          32768;
  4:       712909,       4162176,       9806400,       11816960,        7557120,        2359296,         262144;
  5:      7863667,      53331656,     150752448,      228569600,      198553600,       97026048,       23855104,        2097152;
  6:     87615745,     676981760,    2232810496,     4083236864,     4492902400,     3009413120,     1174405120,      234881024,      16777216;
  7:    983726695,    8530832520,   32162780160,    68896235520,    91866120192,    78405304320,    42278584320,    13589544960,    2264924160,     134217728;
  8:  11112210781,  106872295040,  453406040640,  1113564856320,  1745348935680,  1812760363008,  1252172759040,   560191242240,  151749918720,   21474836480,   1073741824;
  9: 126142119187, 1332489936008, 6282753400000, 17404748444160, 31361590149120, 38409689235456, 32411357282304, 18664862515200, 7083927797760, 1646180433920, 200789721088, 8589934592;

The row polynomials satisfy (n+2)*P(n,x) = -(25*n+25+(80*n+80)*x+(64*n+64)*x^2)*P(n-2,x) + (14*n+21+(16*n+24)*x)*P(n-1,x) for n >= 2, with P(0,x) = 61 + 128*x + 64*x^2, P(1,x) = 595 + 1800*x + 1728*x^2 + 512*x^3.

P[0, x_] := 61 + 128*x + 64*x^2;
P[1, x_] := 595 + 1800*x + 1728*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(25*n + 25 + (80*n + 80)*x + (64*n + 64)*x^2)*P[n - 2, x] + (14*n + 21 + (16*n + 24)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 61 + 128*x + 64*x^2; v[2] = 595 + 1800*x + 1728*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(25*n + 25 + (80*n + 80)*x + (64*n + 64)*x^2)*v[i - 2] + (14*n + 21 + (16*n + 24)*x)*v[i - 1])/(n + 2)); 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)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of Delannoy paths counted by number of weak peaks.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Delannoy paths counted by number of weak peaks.?
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Delannoy paths counted by number of weak peaks.?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A104684 → basis A013611 → basis A013609

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
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.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023