A038331 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:             510,              450;
  1:           13005,            22950,             10125;
  2:          294780,           780300,            688500,            202500;
  3:         6264075,         22108500,          29261250,          17212500,           3796875;
  4:       127787130,        563766750,         994882500,         877837500,         387281250,          68343750;
  5:      2534444745,      13417648650,       29597754375,       34820887500,       23043234375,        8132906250,        1196015625;
  6:     49240640760,     304133369400,      805058919000,     1183910175000,     1044626625000,      553037625000,      162658125000,       20503125000;
  7:    941727254535,    6647486502600,    20529002434500,    36227651355000,    39956968406250,    28204918875000,    12443346562500,     3136978125000,     345990234375;
  8:  17788181474550,  141259088180250,   498561487695000,  1026450121725000,  1358536925812500,  1198709052187500,   705122971875000,   266643140625000,   58818339843750,    5766503906250;
  9: 332638993574085, 2935049943300750, 11653874774870625, 27420881823225000, 42341067521156250, 44831718551812500, 32964498935156250, 16620755765625000, 5499514775390625, 1078336230468750, 95147314453125;

The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(15*n+30)*x)*P(n-1,x) for n >= 1, with P(0,x) = 510 + 450*x.

P[0, x_] := 510 + 450*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (15*n + 30)*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] = 510 + 450*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (15*n + 30)*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)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
        • 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

Related ?

Same fingerprint ?

• A147716 → basis A013609 → x ↦ x/(x+1)
• A038331 → basis A013610 → reverse
• A038331 → basis A013610
• A038271 → basis A007318 → basis A013610

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.