Real-rooted atlas

A266213 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(168 + 1026\,x + 1566\,x^{2} - 162\,x^{3} - 486\,x^{4} - 72\,n - 576\,n x - 2160\,n x^{2} - 486\,n x^{3} + 243\,n x^{4} - 270\,n^{2} x + 648\,n^{2} x^{2} + 243\,n^{2} x^{3} + 162\,n^{3} x\right)\,P_{n}(x) = \left(-168 - 1026\,x - 1566\,x^{2} + 162\,x^{3} + 486\,x^{4} - 96\,n + 774\,n x + 2916\,n x^{2} + 162\,n x^{3} - 729\,n x^{4} + 576\,n^{2} + 1080\,n^{2} x - 1296\,n^{2} x^{2} - 729\,n^{2} x^{3} - 216\,n^{3} - 486\,n^{3} x\right)\,P_{n-1}(x) + \left(-32 - 168\,x + 252\,x^{2} + 2943\,x^{3} + 5751\,x^{4} + 3645\,x^{5} + 729\,x^{6} + 48\,n + 864\,n x + 4158\,n x^{2} + 7290\,n x^{3} + 4131\,n x^{4} + 729\,n x^{5} + 144\,n^{2} + 864\,n^{2} x + 1458\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
29615
3316018109
410133628854779
53051452221411881625321
688555021224011718453648635367
7249519176564127948854756165241458230309
86889631202315524344844416122361965832043741476547
918721199272876996206344828026002211300963738201204131229350401
10502416094143130848887619615166764160127281028181637747626823443936658624479
111334711817604106814703547497673250892974673008375772645227160143292242283228118098364541149
123516295312448351570241339480263248142125232324605671386724096644661905576845306536875582723542942251154555
13919889152699401123946824833886681347590034255367825233545591103056997348190073482277574639619261783824800580106288213819760441
1423921734328067035076024016812432905300541180115655368861789248852019771551630155066610608451683370307054800068768465480779032318864684408339351
Technical details
Node id
n-635dafcb5fe8fafe · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5a963a91e97334a6

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:     1;
  2:     9,      6;
  3:    31,     60,     18;
  4:   101,    336,    288,      54;
  5:   305,   1452,   2214,    1188,     162;
  6:   885,   5502,  12240,   11718,    4536,     486;
  7:  2495,  19176,  56412,   79488,   54756,   16524,   1458;
  8:  6889,  63120, 231552,  434484,  441612,  236196,  58320,   4374;
  9: 18721, 199272, 876996, 2063448, 2802600, 2211300, 963738, 201204, 13122;
%F Formula
The row polynomials satisfy -(72*n-168-(162*n^3-270*n^2-576*n+1026)*x-(648*n^2-2160*n+1566)*x^2-(243*n^2-486*n-162)*x^3-(243*n-486)*x^4)*P(n,x) = -(216*n^3-576*n^2+96*n+168+(486*n^3-1080*n^2-774*n+1026)*x+(1296*n^2-2916*n+1566)*x^2+(729*n^2-162*n-162)*x^3+(729*n-486)*x^4)*P(n-1,x) + (144*n^2+48*n-32+(864*n^2+864*n-168)*x+(1458*n^2+4158*n+252)*x^2+(486*n^2+7290*n+2943)*x^3+(4131*n+5751)*x^4+(729*n+3645)*x^5+729*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[(-(216*n^3 - 576*n^2 + 96*n + 168 + (486*n^3 - 1080*n^2 - 774*n + 1026)*x + (1296*n^2 - 2916*n + 1566)*x^2 + (729*n^2 - 162*n - 162)*x^3 + (729*n - 486)*x^4)*P[n - 1, x] + (144*n^2 + 48*n - 32 + (864*n^2 + 864*n - 168)*x + (1458*n^2 + 4158*n + 252)*x^2 + (486*n^2 + 7290*n + 2943)*x^3 + (4131*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(-72*n + 168 + (162*n^3 - 270*n^2 - 576*n + 1026)*x + (648*n^2 - 2160*n + 1566)*x^2 + (243*n^2 - 486*n - 162)*x^3 + (243*n - 486)*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] = (-(216*n^3 - 576*n^2 + 96*n + 168 + (486*n^3 - 1080*n^2 - 774*n + 1026)*x + (1296*n^2 - 2916*n + 1566)*x^2 + (729*n^2 - 162*n - 162)*x^3 + (729*n - 486)*x^4)*v[i - 1] + (144*n^2 + 48*n - 32 + (864*n^2 + 864*n - 168)*x + (1458*n^2 + 4158*n + 252)*x^2 + (486*n^2 + 7290*n + 2943)*x^3 + (4131*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(-72*n + 168 + (162*n^3 - 270*n^2 - 576*n + 1026)*x + (648*n^2 - 2160*n + 1566)*x^2 + (243*n^2 - 486*n - 162)*x^3 + (243*n - 486)*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.