Real-rooted atlas

A132166 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(3 + 84\,x + n + 42\,n x\right)\,P_{n-1}(x) + \left(-x - 36\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 42\,x + 336\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0142336379
1163111345365713
21842331231845443280032
311053990652054122725987521080325
41126609013986015180486531840615859214354557
5114786312564104010769309128409461836860046272189853438
611681161342411687239259930211256736201612749218565604318722511177679
7118915036652239166854872537909012193126624957374121616176101760504388684833258000301

All 15 rows

Technical details
Node id
n-ab60eb3242173f47 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f34dcee26c4e5272

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,  42,   336;
  1: 1,  63,  1113,    4536;
  2: 1,  84,  2331,   23184,    54432;
  3: 1, 105,  3990,   65205,   412272,     598752;
  4: 1, 126,  6090,  139860,  1518048,    6531840,     6158592;
  5: 1, 147,  8631,  256410,  4010769,   30912840,    94618368,     60046272;
  6: 1, 168, 11613,  424116,  8723925,   99302112,   567362016,   1274921856,    560431872;
  7: 1, 189, 15036,  652239, 16685487,  253790901,  2193126624,   9573741216,  16176101760,    5043886848;
  8: 1, 210, 18900,  950040, 29117907,  557085438,  6534131760,  44153890560, 150628802688,  195030291456,   44019376128;
  9: 1, 231, 23205, 1326780, 47438118, 1096591041, 16358573673, 152347155480, 822902685696, 2232875211840, 2250184914432, 374164697088;
%F Formula
The row polynomials satisfy (n+3)*P(n,x) = (n+3+(42*n+84)*x)*P(n-1,x) - (x+36*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 42*x + 336*x^2.
%t Mathematica
P[0, x_] := 1 + 42*x + 336*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 3 + (42*n + 84)*x)*P[n - 1, x] - (x + 36*x^2)*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 42*x + 336*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 3 + (42*n + 84)*x)*v[i - 1] - (x + 36*x^2)*deriv(v[i - 1], x))/(n + 3)); 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. [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