A038320 → D

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Submission text ?

Triangle begins:
  0:           132,            72;
  1:          2178,          2376,            648;
  2:         31944,         52272,          28512,           5184;
  3:        439230,        958320,         784080,         285120,          38880;
  4:       5797836,      15812280,       17249760,        9408960,        2566080,         279936;
  5:      74405562,     243509112,      332057880,      241496640,       98794080,       21555072,       1959552;
  6:     935384208,    3571466976,     5844218688,     5312926080,     2897959680,      948423168,     172440576,      13436928;
  7:   11575379574,   50510747232,    96429608352,   105195936384,    71724502080,    31297964544,    8535808512,    1330255872,     90699264;
  8:  141476861460,  694522774440,  1515322416960,  1928592167040,  1577939045760,   860694024960,  312979645440,   73164072960,   9976919040,   604661760;
  9: 1711870023666, 9337472856360, 22919251556520, 33337093173120, 31821770756160, 20828795404032, 9467634274560, 2950950942720, 603603601920, 73164072960, 3990767616;

The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(6*n+12)*x)*P(n-1,x) for n >= 1, with P(0,x) = 132 + 72*x.

P[0, x_] := 132 + 72*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (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] = 132 + 72*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (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 Triangle whose (i,j)-th entry is binomial(i,j)*11^(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
    • Geometric basis map (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) [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) (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)
    • 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 ?

• A038271 → basis A013609 → x ↦ x/(x+1)
• A027466 → basis A013610 → basis A013609
• A038314 → x ↦ x+1 → D
• A038336 → x ↦ x/(x+1)
• A013612 → x ↦ x+1 → x ↦ x/(x+1)
• A038259 → x ↦ x/(x+1)
• A038334 → basis A013610 → basis A007318
• A038320 → reverse
• A038314 → x ↦ x+1
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• A038277 → basis A013610
• A038336 → x ↦ x/(x+1) → 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.