A038305 → basis A013609 → D

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Submission text ?

Triangle begins:
  0:           156,             72;
  1:          3042,           2808,            648;
  2:         52728,          73008,          33696,            5184;
  3:        856830,        1581840,        1095120,          336960,          38880;
  4:      13366548,       30845880,       28473120,        13141440,        3032640,         279936;
  5:     202725978,      561395016,      647763480,       398623680,      137985120,       25474176,        1959552;
  6:    3011928816,     9730846944,    13473480384,     10364215680,     4783484160,     1324657152,      203793408,      13436928;
  7:   44049458934,   162644156064,   262732867488,    242522646912,   139916911680,    51661628928,    11921914368,    1572120576,     90699264;
  8:  636269962380,  2642967536040,  4879324681920,   5254657349760,  3637839703680,  1679002940160,   516616289280,  102187837440,  11790904320,   604661760;
  9: 9098660462034, 41993817517080, 87217928689320, 107345143002240, 86701846271040, 48019484088576, 18469032341760, 4870953584640, 843049658880, 86466631680, 3990767616;

The row polynomials satisfy (n+1)*P(n,x) = (13*n+26+(6*n+12)*x)*P(n-1,x) for n >= 1, with P(0,x) = 156 + 72*x.

P[0, x_] := 156 + 72*x;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + 26 + (6*n + 12)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 156 + 72*x; for(n = 1, N, my(i = n + 1); v[i] = ((13*n + 26 + (6*n + 12)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(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
        • 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

Related ?

Same fingerprint ?

• A038284 → basis A013609 → basis A007318
• A038291 → basis A013610 → basis A013609
• A038305 → basis A013609
• A038335 → basis A013609 → x ↦ x+1
• A038305 → basis A013609 → reverse
• A038321 → basis A013610 → x ↦ x/(x+1)

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.