A038330 → basis A007318 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:            384,             160,              16;
  1:           7168,            4608,             960,              64;
  2:         131072,          114688,           36864,            5120,            256;
  3:        2359296,         2621440,         1146880,          245760,          25600,           1024;
  4:       41943040,        56623104,        31457280,         9175040,        1474560,         122880,          4096;
  5:      738197504,      1174405120,       792723456,       293601280,       64225280,        8257536,        573440,        16384;
  6:    12884901888,     23622320128,     18790481920,      8455716864,     2348810240,      411041792,      44040192,      2621440,       65536;
  7:   223338299392,    463856467968,    425201762304,    225485783040,    76101451776,    16911433728,    2466250752,    226492416,    11796480,     262144;
  8:  3848290697216,   8933531975680,   9277129359360,   5669356830720,  2254857830400,   608811614208,  112742891520,  14092861440,  1132462080,   52428800,   1048576;
  9: 65970697666560, 169324790677504, 196537703464960, 136064563937280, 62362925137920, 19842748907520, 4464618504192, 708669603840, 77510737920, 5536481280, 230686720, 4194304;

The row polynomials satisfy (n+5+x)*P(n,x) = (16*n+96+(4*n+40)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 384 + 160*x + 16*x^2.

P[0, x_] := 384 + 160*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 96 + (4*n + 40)*x + 4*x^2)*P[n - 1, x])/(n + 5 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 384 + 160*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 96 + (4*n + 40)*x + 4*x^2)*v[i - 1])/(n + 5 + 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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038231 → D+1
• A038231 → D → D+1
• A027467 → basis A013611 → D+1
• A038329 → D+1
• A027466 → basis A013609 → 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.