A038284 → basis A013609 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:            532,             624,              144;
  1:           9800,           19152,            11232,             1728;
  2:         170128,          470400,           459648,           179712,            20736;
  3:        2842784,        10207680,         14112000,          9192960,          2695680,           248832;
  4:       46252864,       204680448,        367476480,        338688000,        165473280,         38817792,          2985984;
  5:      737894528,      3885240576,       8596578816,      10289341440,       7112448000,       2779951104,        543449088,         35831808;
  6:    11595485440,     70837874688,     186491547648,     275090522112,     246944194560,     136559001600,      44479217664,       7453016064,       429981696;
  7:   180046264832,   1252312427520,    3825245233152,    6713695715328,    7427444097024,    5333994602496,    2458062028800,     686250786816,    100615716864,      5159780352;
  8:  2768580269056,  21605551779840,   75138745651200,  153009809326080,  201410871459840,  178258658328576,  106679892049920,   42138206208000,  10293761802240,   1341542891520,    61917364224;
  9: 42231179626496, 365452595515392, 1425966417469440, 3306104808652800, 5049323707760640, 5317247006539776, 3921690483228672, 2011677964369920, 695280402432000, 150975173099520, 17708366168064, 743008370688;

The row polynomials satisfy (6*n+13+6*x)*P(n,x) = (84*n+266+(72*n+312)*x+72*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 532 + 624*x + 144*x^2.

P[0, x_] := 532 + 624*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((84*n + 266 + (72*n + 312)*x + 72*x^2)*P[n - 1, x])/(6*n + 13 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 532 + 624*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((84*n + 266 + (72*n + 312)*x + 72*x^2)*v[i - 1])/(6*n + 13 + 6*x)); 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)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(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
        • 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 ?

• A038335 → basis A013609 → D+1
• A013613 → x ↦ x+1 → D+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.