A038334 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:              960,              1152;
  1:            28800,             69120,              41472;
  2:           768000,           2764800,            3317760,            1327104;
  3:         19200000,          92160000,          165888000,          132710400,            39813120;
  4:        460800000,        2764800000,         6635520000,         7962624000,          4777574400,          1146617856;
  5:      10752000000,       77414400000,       232243200000,       371589120000,        334430208000,        160526499840,         32105299968;
  6:     245760000000,     2064384000000,      7431782400000,     14863564800000,      17836277760000,      12842119987200,       5136847994880,        880602513408;
  7:    5529600000000,    53084160000000,    222953472000000,    535088332800000,     802632499200000,     770527199232000,     462316319539200,     158508452413440,     23776267862016;
  8:  122880000000000,  1327104000000000,   6370099200000000,  17836277760000000,   32105299968000000,   38526359961600000,   30821087969280000,   15850845241344000,   4755253572403200,    634033809653760;
  9: 2703360000000000, 32440320000000000, 175177728000000000, 560568729600000000, 1177194332160000000, 1695159838310400000, 1695159838310400000, 1162395317698560000, 523077892964352000, 139487438123827200, 16738492574859264;

The row polynomials satisfy (n+1)*P(n,x) = (20*n+40+(24*n+48)*x)*P(n-1,x) for n >= 1, with P(0,x) = 960 + 1152*x.

P[0, x_] := 960 + 1152*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (24*n + 48)*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] = 960 + 1152*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n + 40 + (24*n + 48)*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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
        • 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 ?

• A120910 → basis A013609 → basis A013610
• A038291 → basis A013609 → x ↦ x/(x+1)
• A038220 → basis A013610 → D
• A038334 → basis A013610
• A038255 → basis A013610 → basis A013611
• A038259 → D
• A147716 → basis A013610 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A038221 → basis A013610 → basis A013609
• A038314 → D
• A178820 → basis A013610 → basis A013609
• A038220 → basis A013610
• A178821 → basis A013610 → basis A013609
• A038329 → x ↦ x+1 → x ↦ x/(x+1)
• A013612 → x ↦ x+1 → reverse
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A038336 → D
• A013612 → x ↦ x+1 → D
• A013612 → x ↦ x+1
• A038334 → basis A013610 → reverse
• A178822 → basis A013610 → basis A013609
• A038271 → basis A013609
• A038338 → basis A013610 → basis A013609
• A038271 → basis A013609 → D
• A196347 → basis A013610 → basis A013609
• A028326 → basis A013610 → basis A013609
• A085880 → basis A013610 → basis A013609
• A156992 → basis A013610 → basis A013609

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.