A038330 → basis A013609 → basis A008277

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:              576,               448,                 64;
  1:            13824,             18944,               6144,                512;
  2:           331776,            716800,             397312,              73728,               4096;
  3:          7962624,          25591808,           21626880,            6717440,             819200,              32768;
  4:        191102976,         882638848,         1069809664,          495452160,           99614720,            8650752,            262144;
  5:       4586471424,       29773266944,        49853497344,        32340180992,         9542041600,         1350565888,          88080384,          2097152;
  6:     110075314176,      989436313600,      2232326029312,      1951156666368,       793075777536,       162772549632,       17146314752,        872415232,        16777216;
  7:    2641807540224,    32542564548608,     97208531681280,    111514128220160,     60021496872960,     16762049396736,     2536715059200,     206963736576,      8455716864,      134217728;
  8:   63403380965376,  1062496525877248,   4148681783640064,   6130346408017920,   4253316850647040,   1552943136374784,   316739879436288,   36850819399680,   2399812976640,    80530636800,   1073741824;
  9: 1521681143169024, 34507115875794944, 174447243552620544, 327446081853980672, 287228514900377600, 133414895533162496, 35228816410411008, 5481984587464704, 505990097141760, 26929444945920, 755914244096, 8589934592;

The row polynomials satisfy P(n,x) = (24+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 576 + 448*x + 64*x^2.

P[0, x_] := 576 + 448*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 8*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 448*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 8*x)*v[i - 1] + 8*x*deriv(v[i - 1], 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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 ?

• A085880 → x ↦ x+1 → basis A008277
• A038207 → basis A008277 → reverse
• A143495 → reverse
• basis-map coefficients of A113214, shifted → x ↦ x+1 → basis A008277
• A156992 → basis A007318 → basis A008277
• A178821 → basis A007318 → basis A008277
• A038208 → basis A007318 → basis A008277
• A038338 → x ↦ x+1 → basis A008277
• A135065 → x ↦ x+1 → basis A008277
• A028326 → basis A007318 → basis A008277
• A094305 → x ↦ x+1 → basis A008277
• A356546 → x ↦ x+1 → basis A008277
• A038325 → basis A007318 → basis A008277
• A196347 → x ↦ x+1 → basis A008277
• A120910 → basis A008277
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
• A085881 → basis A007318 → basis A008277
• A003506 → x ↦ x+1 → basis A008277
• A027466 → basis A013611 → basis A008277
• A178822 → x ↦ x+1 → basis A008277
• A178820 → basis A007318 → basis A008277
• A038207 → basis A008277
• A120910 → D → basis A008277
• A038221 → x ↦ x+1 → basis A008277

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.