A038330 → basis A013609 → θ+1

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Triangle begins:
  0:            256,             512,             192;
  1:           4096,           12288,            9216,             2048;
  2:          65536,          262144,          294912,           131072,            20480;
  3:        1048576,         5242880,         7864320,          5242880,          1638400,           196608;
  4:       16777216,       100663296,       188743680,        167772160,         78643200,         18874368,         1835008;
  5:      268435456,      1879048192,      4227858432,       4697620480,       2936012800,       1056964608,       205520896,        16777216;
  6:     4294967296,     34359738368,     90194313216,     120259084288,      93952409600,      45097156608,     13153337344,      2147483648,       150994944;
  7:    68719476736,    618475290624,   1855425871872,    2886218022912,    2705829396480,    1623497637888,    631360192512,    154618822656,     21743271936,     1342177280;
  8:  1099511627776,  10995116277760,  37108517437440,   65970697666560,   72155450572800,   51951924412416,  25254407700480,   8246337208320,   1739461754880,   214748364800,   11811160064;
  9: 17592186044416, 193514046488576, 725677674332160, 1451355348664320, 1814194185830400, 1523923116097536, 888955151056896, 362838837166080, 102048422952960, 18897856102400, 2078764171264, 103079215104;

The row polynomials satisfy (2+(n+2)*x)*P(n,x) = (32+(16*n+64)*x+(8*n+24)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 512*x + 192*x^2.

P[0, x_] := 256 + 512*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((32 + (16*n + 64)*x + (8*n + 24)*x^2)*P[n - 1, x])/(2 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 512*x + 192*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((32 + (16*n + 64)*x + (8*n + 24)*x^2)*v[i - 1])/(2 + (n + 2)*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)*12^(i-j)*4^j.?
    • 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

Related ?

Same fingerprint ?

• A003506 → x ↦ x+1 → θ+1
• A120910 → D → θ+1
• A028326 → basis A007318 → θ+1
• A085880 → x ↦ x+1 → θ+1
• A120910 → θ+1
• image under θ+1 → θ+1 → D+1 → basis A013609
• A038338 → x ↦ x+1 → θ+1
• A038221 → x ↦ x+1 → θ+1
• A156992 → basis A007318 → θ+1
• basis-map coefficients of A113214, shifted → x ↦ x+1 → θ+1
• A196347 → x ↦ x+1 → θ+1
• A132440 → D+1 → basis A013609
• A038325 → basis A007318 → θ+1
• A038207 → θ+1
• A135065 → x ↦ x+1 → θ+1
• A038208 → basis A007318 → θ+1
• A094305 → x ↦ x+1 → θ+1
• A178821 → basis A007318 → θ+1
• A356546 → x ↦ x+1 → θ+1
• A027466 → basis A013611 → θ+1
• A178820 → basis A007318 → θ+1
• A085881 → basis A007318 → θ+1
• A131431 → D+1 → basis A013609
• A178822 → x ↦ x+1 → 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.