Real-rooted atlas

A272866 → D → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 20\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
15813
2502804802561066
33753600132002304019200614465559
4250034000192000582400102400010444805734401310723583892
5156252750002100000912000024800000437760005017600036044800147456002621440183674465
69375020250001950000011040000040800000010329600001827840000226099200019169280001061683200346030080503316489036783678
75468751400000016275000011368000005320000000176064000004238080000075120640000980582400009312665600062573772800281857228807633633280939524096432259485931

All 15 rows

Technical details
Node id
n-877f873b9e80c2e2 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-dc1f92b0a0065af1

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,         8;
  2:       50,       280,        480,         256;
  3:      375,      3600,      13200,       23040,        19200,          6144;
  4:     2500,     34000,     192000,      582400,      1024000,       1044480,         573440,         131072;
  5:    15625,    275000,    2100000,     9120000,     24800000,      43776000,       50176000,       36044800,       14745600,         2621440;
  6:    93750,   2025000,   19500000,   110400000,    408000000,    1032960000,     1827840000,     2260992000,     1916928000,      1061683200,       346030080,        50331648;
  7:   546875,  14000000,  162750000,  1136800000,   5320000000,   17606400000,    42380800000,    75120640000,    98058240000,     93126656000,     62573772800,     28185722880,      7633633280,       939524096;
  8:  3125000,  92500000, 1260000000, 10472000000,  59360000000,  242995200000,   741888000000,  1719910400000,  3052339200000,   4147118080000,   4279238656000,   3294206361600,   1832071987200,    695247831040,   161061273600,    17179869184;
  9: 17578125, 590625000, 9225000000, 88920000000, 592200000000, 2890944000000, 10708992000000, 30740889600000, 69221376000000, 122963558400000, 172322979840000, 189460905984000, 161451343872000, 104428103270400, 49526341632000, 16234976378880, 3285649981440, 309237645312;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+20*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 8*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 20*n*x + 16*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 20*n*x + 16*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.