A038321 → basis A013610 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:             756,              882;
  1:           20412,            47628,             27783;
  2:          489888,          1714608,           2000376,             777924;
  3:        11022480,         51438240,          90016920,           70013160,           20420505;
  4:       238085568,       1388832480,        3240609120,         3780710640,         2205414540,          514596726;
  5:      4999796928,      34998578496,      102079187280,       158789846880,       138941116020,        64839187476,        12607619787;
  6:    102852965376,     839965883904,     2939880593664,      5716434487680,      6669173568960,      4668421498272,      1815497249328,       302582874888;
  7:   2082772548864,   19439210456064,    79376776028928,    185212477400832,    270101529542880,    252094760906688,    147055277195568,     49018425731856,      7148520419229;
  8:  41655450977280,  437382235261440,  2041117097886720,   5556374322024960,   9723655063543680,  11344264240800960,   8823316631734080,   4411658315867040,   1286733675461220,   166798809782010;
  9: 824777929350144, 9622409175751680, 50517648172696320, 157166016537277440, 320880617096941440, 449232863935718016, 436754173270836960, 291169448847224640, 127386633870660780, 33026164336837980, 3853052505964431;

The row polynomials satisfy (n+1)*P(n,x) = (18*n+36+(21*n+42)*x)*P(n-1,x) for n >= 1, with P(0,x) = 756 + 882*x.

P[0, x_] := 756 + 882*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 36 + (21*n + 42)*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] = 756 + 882*x; for(n = 1, N, my(i = n + 1); v[i] = ((18*n + 36 + (21*n + 42)*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)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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
        • 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 ?

• A038284 → basis A013609 → D
• A038291 → basis A013611 → basis A013610
• A038284 → basis A013609 → reverse
• A038335 → basis A013609 → reverse
• A013613 → x ↦ x+1
• A038335 → basis A013609
• A038284 → basis A013609
• A038321 → basis A013610
• A013613 → x ↦ x+1 → reverse
• A038329 → basis A013609 → basis A013610
• A038335 → basis A013609 → D
• A013613 → x ↦ x+1 → D
• A038330 → basis A013610 → basis A013609
• A038328 → x ↦ x/(x+1)
• A038255 → 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.