Real-rooted atlas

A269951 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20 + 9\,n - n^{2}\right)\,P_{n-4}(x) + \left(20 - 8\,x - 13\,n + 2\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(3 + 4\,x - x^{2} + 2\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 2 + 2\,x\)
\(P_{4}(x) = 5 + 10\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
3224
4510318
5164627493
665232195565553
73261338141058010063760
8195788581090256601400162728946
9137006675092043563361767529402458249697
1010960156614485219859012821969546536558635292390249
11986410534664286348526586856279321070547410819298404861025171972
129864101556905869533355378657400369816251069324219628912287201633565011289429114
131085051126345488141141189995100513164451416915016564211434948221488954444847025850847123609499773
1413020613457853548360147386240731372306663275338156302652724800625984205100988976111511628272003932510801348542832801
15169267974861049392139622044267796221997128518841164572223104419102058811439245818207058165626434722623660780145030657876135214700453230880
Technical details
Node id
n-e81c2862c7b1d09a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d90da0c261762bff

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      0;
   2:      1;
   3:      2,      2;
   4:      5,     10,      3;
   5:     16,     46,     27,      4;
   6:     65,    232,    195,     56,      5;
   7:    326,   1338,   1410,    580,    100,     6;
   8:   1957,   8858,  10902,   5660,   1400,   162,    7;
   9:  13700,  66750,  92043,  56336,  17675,  2940,  245,   8;
  10: 109601, 566144, 852198, 590128, 219695, 46536, 5586, 352, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-9*n+20)*P(n-4,x) + (2*n^2-13*n+20+(2*n-8)*x)*P(n-3,x) - (n^2-2*n-3+(2*n-4)*x+x^2)*P(n-2,x) + (2*n-3+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 2 + 2*x, P(4,x) = 5 + 10*x + 3*x^2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 2 + 2*x;
P[4, x_] := 5 + 10*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 9*n + 20)*P[n - 4, x] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*P[n - 3, x] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*P[n - 2, x] + (2*n - 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 2 + 2*x; v[4] = 5 + 10*x + 3*x^2; for(n = 5, N, my(i = n); v[i] = -(n^2 - 9*n + 20)*v[i - 4] + (2*n^2 - 13*n + 20 + (2*n - 8)*x)*v[i - 3] - (n^2 - 2*n - 3 + (2*n - 4)*x + x^2)*v[i - 2] + (2*n - 3 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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