A038328 → basis A013609 → basis A013609

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:            324,             288,              64;
  1:           5832,            7776,            3456,             512;
  2:         104976,          186624,          124416,           36864,            4096;
  3:        1889568,         4199040,         3732480,         1658880,          368640,           32768;
  4:       34012224,        90699264,       100776960,        59719680,        19906560,         3538944,          262144;
  5:      612220032,      1904684544,      2539579392,      1881169920,       836075520,       222953472,        33030144,        2097152;
  6:    11019960576,     39182082048,     60949905408,     54177693696,     30098718720,     10701766656,      2378170368,      301989888,       16777216;
  7:   198359290368,    793437161472,   1410554953728,   1462797729792,    975198486528,    433421549568,    128421199872,    24461180928,     2717908992,     134217728;
  8:  3570467226624,  15868743229440,  31737486458880,  37614798766080,  29255954595840,  15603175784448,   5778953994240,  1467670855680,   244611809280,   24159191040,   1073741824;
  9: 64268410079232, 314201115942912, 698224702095360, 930966269460480, 827525572853760, 514904800886784, 228846578171904, 72649707356160, 16144379412480, 2391759912960, 212600881152, 8589934592;

The row polynomials satisfy P(n,x) = (18+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 288*x + 64*x^2.

P[0, x_] := 324 + 288*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 288*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 8*x)*v[i - 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 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

Related ?

Same fingerprint ?

• A013610 → basis A013610
• A038329 → basis A013611 → x ↦ x+1
• A013610 → basis A013610 → D
• A038319 → basis A013611 → x ↦ x/(x+1)
• A038279 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013610
• A038334 → basis A013609 → basis A007318
• A130123 → basis A013610 → basis A013610
• A038279 → basis A013611 → D
• A136572 → basis A013610 → basis A013610
• A038271 → basis A013610 → x ↦ x/(x+1)
• A013610 → D → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013610
• A013610 → basis A013611 → basis A013610
• A013622 → basis A013609 → 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.