Real-rooted atlas

A249250 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 3\,x - 11\,x^{2} - 11\,x^{3} - 2\,x^{4} - 3\,n - 15\,n x - 10\,n x^{2} + 7\,n x^{3} + 2\,n x^{4} + 6\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{2} x^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(3 + 3\,x - 11\,x^{2} - 11\,x^{3} - 2\,x^{4} + 15\,n + 21\,n x - 43\,n x^{2} - 53\,n x^{3} - 10\,n x^{4} - 9\,n^{2} - 39\,n^{2} x - 54\,n^{2} x^{2} - 10\,n^{2} x^{3} - 9\,n^{3} - 15\,n^{3} x\right)\,P_{n-1}(x) + \left(18 + 123\,x + 321\,x^{2} + 395\,x^{3} + 227\,x^{4} + 52\,x^{5} + 4\,x^{6} + 27\,n + 144\,n x + 267\,n x^{2} + 198\,n x^{3} + 50\,n x^{4} + 4\,n x^{5} + 9\,n^{2} + 36\,n^{2} x + 39\,n^{2} x^{2} + 6\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
234481294
312428816824604
441213441368480483652
5133656408304508812489621712
64216218884252838400164163072192126712
71307280640195264235584149184483847296384729808
83995228569683001612602881074240516096134016168967684157968
91207369823683332736611712066044164284288164428835481638400153623480704
10361504329856012806208276111363623385629887488154982404927488907776860163072131621344
1110741121086412847519616117809280182437632183591936121288704520642561408512022609921904646144733192384
123170752352143361713880324805882888590510081023602688826957824452542464165055488387809285511168417792122884062293056
139308032112614528603733248188992358438334259205288205312507066777633977794561579603968499476480103575552131973129093122457622402445056
14271932163560202242084840448720887808016368995328256909148162858383564822768582656129684172805223284736145492377626974617631137792196608049152123038785408
Technical details
Node id
n-ea06c2ded5f0690b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4635ca94cfd32645

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:     10,      6;
  2:     34,     48,      12;
  3:    124,    288,     168,      24;
  4:    412,   1344,    1368,     480,      48;
  5:   1336,   5640,    8304,    5088,    1248,      96;
  6:   4216,  21888,   42528,   38400,   16416,    3072,     192;
  7:  13072,  80640,  195264,  235584,  149184,   48384,    7296,    384;
  8:  39952, 285696,  830016, 1260288, 1074240,  516096,  134016,  16896,   768;
  9: 120736, 982368, 3332736, 6117120, 6604416, 4284288, 1644288, 354816, 38400, 1536;
%F Formula
The row polynomials satisfy -(3*n-3-(3*n^3+6*n^2-15*n+3)*x-(12*n^2-10*n-11)*x^2-(2*n^2+7*n-11)*x^3-(2*n-2)*x^4)*P(n,x) = -(9*n^3+9*n^2-15*n-3+(15*n^3+39*n^2-21*n-3)*x+(54*n^2+43*n+11)*x^2+(10*n^2+53*n+11)*x^3+(10*n+2)*x^4)*P(n-1,x) + (9*n^2+27*n+18+(36*n^2+144*n+123)*x+(39*n^2+267*n+321)*x^2+(6*n^2+198*n+395)*x^3+(50*n+227)*x^4+(4*n+52)*x^5+4*x^6)*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[Together[(-(9*n^3 + 9*n^2 - 15*n - 3 + (15*n^3 + 39*n^2 - 21*n - 3)*x + (54*n^2 + 43*n + 11)*x^2 + (10*n^2 + 53*n + 11)*x^3 + (10*n + 2)*x^4)*P[n - 1, x] + (9*n^2 + 27*n + 18 + (36*n^2 + 144*n + 123)*x + (39*n^2 + 267*n + 321)*x^2 + (6*n^2 + 198*n + 395)*x^3 + (50*n + 227)*x^4 + (4*n + 52)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-3*n + 3 + (3*n^3 + 6*n^2 - 15*n + 3)*x + (12*n^2 - 10*n - 11)*x^2 + (2*n^2 + 7*n - 11)*x^3 + (2*n - 2)*x^4)]];
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] = (-(9*n^3 + 9*n^2 - 15*n - 3 + (15*n^3 + 39*n^2 - 21*n - 3)*x + (54*n^2 + 43*n + 11)*x^2 + (10*n^2 + 53*n + 11)*x^3 + (10*n + 2)*x^4)*v[i - 1] + (9*n^2 + 27*n + 18 + (36*n^2 + 144*n + 123)*x + (39*n^2 + 267*n + 321)*x^2 + (6*n^2 + 198*n + 395)*x^3 + (50*n + 227)*x^4 + (4*n + 52)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(-3*n + 3 + (3*n^3 + 6*n^2 - 15*n + 3)*x + (12*n^2 - 10*n - 11)*x^2 + (2*n^2 + 7*n - 11)*x^3 + (2*n - 2)*x^4)); 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

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.