A038284 → basis A013609 → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           196,            672,             432;
  1:          2744,          14112,           18144,             6912;
  2:         38416,         263424,          508032,           387072,           103680;
  3:        537824,        4609920,        11854080,         13547520,          7257600,          1492992;
  4:       7529536,       77446656,       248935680,        379330560,        304819200,        125411328,         20901888;
  5:     105413504,     1264962048,      4879139328,       9293598720,       9957427200,       6145155072,       2048385024,        286654464;
  6:    1475789056,    20239392768,     91077267456,     208176611328,     278807961600,     229419122688,     114709561344,      32105299968,       3869835264;
  7:   20661046784,   318770436096,   1639390814208,    4371708837888,    7025960632320,    7226702364672,    4817801576448,    2022633897984,     487599243264,     51597803520;
  8:  289254654976,  4958651228160,  28689339248640,   87434176757760,  163939081420800,  202347666210816,  168623055175680,   94389581905920,   34131947028480,   7223692492800,    681091006464;
  9: 4049565169664, 76363228913664, 490906471587840, 1683107902586880, 3606659791257600, 5193590099410944, 5193590099410944, 3633998903377920, 1752106614128640, 556224321945600, 104888014995456, 8916100448256;

The row polynomials satisfy (7+(6*n+12)*x)*P(n,x) = (98+(84*n+336)*x+(72*n+216)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 672*x + 432*x^2.

P[0, x_] := 196 + 672*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (84*n + 336)*x + (72*n + 216)*x^2)*P[n - 1, x])/(7 + (6*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 672*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (84*n + 336)*x + (72*n + 216)*x^2)*v[i - 1])/(7 + (6*n + 12)*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 ?

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