Real-rooted atlas

A272866 → D → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 15\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
15611
250210270108638
33752700742597206075145827753
4250025500108000245700324000247860102060174961073116
5156252062501181250384750078468751038825089302504811400147622519683038900455
69375015187501096875046575000129093750245126250325316250301806000191909250797161501948617021257641353735834
754687510500000915468754795875001683281250417808125075428718751002739500098168962506992385750352374907511904278402418056552232052245801395717

All 15 rows

Technical details
Node id
n-57c3bb526ceab4ae · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-9de613d7fd734864

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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:        0;
  1:        5,         6;
  2:       50,       210,        270,         108;
  3:      375,      2700,       7425,        9720,         6075,         1458;
  4:     2500,     25500,     108000,      245700,       324000,       247860,        102060,         17496;
  5:    15625,    206250,    1181250,     3847500,      7846875,     10388250,       8930250,       4811400,       1476225,        196830;
  6:    93750,   1518750,   10968750,    46575000,    129093750,    245126250,     325316250,     301806000,     191909250,      79716150,      19486170,       2125764;
  7:   546875,  10500000,   91546875,   479587500,   1683281250,   4178081250,    7542871875,   10027395000,    9816896250,    6992385750,    3523749075,    1190427840,     241805655,      22320522;
  8:  3125000,  69375000,  708750000,  4417875000,  18781875000,  57663900000,  132040125000,  229580325000,  305578575000,  311385060000,  240978969000,  139131253800,   58033357200,   16517186280,   2869781400,    229582512;
  9: 17578125, 442968750, 5189062500, 37513125000, 187375781250, 686034562500, 1905970500000, 4103413425000, 6929953734375, 9232680206250, 9704112660000, 8001907137000, 5114189603250, 2480926020300, 882457780500, 216955473840, 32930741565, 2324522934;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+15*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 6*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 15*n*x + 9*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 15*n*x + 9*n*x^2)*v[i - 1])/(n - 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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.