Real-rooted atlas

shift_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(2 - x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 3\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14419
2244834101117
31925766884161322012025
41920768012960120006616219241438143821
52304011520025152031488024918412931244040952012007211137969
63225601935360518784081984008481984602649629961761040112246360377923378138134476597
7516096036126720114831360219340800280691712253854720166667008802539522825401671505281256080143584943626811193741145

All 15 rows

Technical details
Node id
n-59c538f64c72051a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d0b4c8550cf1eaf3

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          4,           4,           1;
  2:         24,          48,          34,           10,            1;
  3:        192,         576,         688,          416,          132,           20,            1;
  4:       1920,        7680,       12960,        12000,         6616,         2192,          414,           38,            1;
  5:      23040,      115200,      251520,       314880,       249184,       129312,        44040,         9520,         1200,          72,           1;
  6:     322560,     1935360,     5187840,      8198400,      8481984,      6026496,      2996176,      1040112,       246360,       37792,        3378,        138,          1;
  7:    5160960,    36126720,   114831360,    219340800,    280691712,    253854720,    166667008,     80253952,     28254016,     7150528,     1256080,     143584,       9436,       268,        1;
  8:   92897280,   743178240,  2732728320,   6123479040,   9344563200,  10276761600,   8403141120,   5192901120,   2439304576,   868063744,   231018592,   44840992,    6080200,    534448,    26422,     526,     1;
  9: 1857945600, 16721510400, 69827788800, 179601408000, 318459617280, 412765839360, 404516444160, 305735086080, 180085287424, 82936793600, 29766314880, 8240939520, 1727332320, 265839264, 28586280, 1974000, 74520, 1040, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+2+4*n*x+(2*n-1)*x^2)*P(n-1,x) - (2*x+3*x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 2 + 4*n*x + (2*n - 1)*x^2)*P[n - 1, x] - (2*x + 3*x^2 + x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 2 + 4*n*x + (2*n - 1)*x^2)*v[i - 1] - (2*x + 3*x^2 + x^3)*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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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