A038284 → basis A008277

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OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:           196,             204,               36;
  1:          2744,            5256,             2160,              216;
  2:         38416,          121584,            87696,            19872,             1296;
  3:        537824,         2662176,          3009600,          1162080,           168480,             7776;
  4:       7529536,        56470464,         94222656,         55244160,         13374720,          1353024,           46656;
  5:     105413504,      1174586496,       2788611840,       2333149056,        839704320,        139781376,        10450944,          279936;
  6:    1475789056,     24124210944,      79551426816,      91392440832,      45907658496,      11188606464,      1361235456,        78382080,        1679616;
  7:   20661046784,    491338953216,    2213082362880,    3401866667520,    2292845667840,     767744635392,    135193411584,     12556809216,      574428672,     10077696;
  8:  289254654976,   9950745345024,   60488175154176,  122138227537920,  107539335383040,   47537837964288,  11366138391552,   1514341785600,   110955432960,   4131855360,    60466176;
  9: 4049565169664, 200750434830336, 1632397026078720, 4271352332138496, 4819324109783040, 2736900882726912, 853533947363328, 152999970342912, 15965287557120, 946698762240, 29265629184, 362797056;

The row polynomials satisfy P(n,x) = (14+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 196 + 204*x + 36*x^2.

P[0, x_] := 196 + 204*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 6*x)*P[n - 1, x] + 6*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] = 196 + 204*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 6*x)*v[i - 1] + 6*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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-ca3a3ab85b442a8e
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-ca3a3ab85b442a8e
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-ca3a3ab85b442a8e

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A008277 e-ca3a3ab85b442a8e

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

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4ef268212a4f2f46 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-4fdd1b3030c1c25c root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A038335 → basis A008277

References

1. [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
2. [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