A038331 → basis A013609 → D

Matches ?

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Submission text ?

Triangle begins:
  0:             340,              200;
  1:            8670,            10200,             3000;
  2:          196520,           346800,           204000,            40000;
  3:         4176050,          9826000,          8670000,          3400000,           500000;
  4:        85191420,        250563000,        294780000,        173400000,         51000000,          6000000;
  5:      1689629830,       5963399400,       8769705000,       6878200000,       3034500000,        714000000,         70000000;
  6:     32827093840,     135170386400,     238535976000,     233858800000,     137564000000,      48552000000,       9520000000,       800000000;
  7:    627818169690,    2954438445600,    6082667388000,    7156079280000,    5261823000000,    2476152000000,     728280000000,    122400000000,      9000000000;
  8:  11858787649700,   62781816969000,  147721922280000,  202755579600000,  178901982000000,  105236460000000,   41269200000000,  10404000000000,   1530000000000,   100000000000;
  9: 221759329049390, 1304466641467000, 3452999933295000, 5416470483600000, 5575778439000000, 3935843604000000, 1929335100000000, 648516000000000, 143055000000000, 18700000000000, 1100000000000;

The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(10*n+20)*x)*P(n-1,x) for n >= 1, with P(0,x) = 340 + 200*x.

P[0, x_] := 340 + 200*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (10*n + 20)*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] = 340 + 200*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (10*n + 20)*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)*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 ?

• A038331 → basis A013609
• A038271 → basis A007318 → basis A013609
• A038331 → basis A013609 → reverse

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.