Real-rooted atlas

A249252 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(62 + 462\,x + 1396\,x^{2} + 2200\,x^{3} + 1914\,x^{4} + 874\,x^{5} + 164\,x^{6} - 100\,n - 722\,n x - 2138\,n x^{2} - 3328\,n x^{3} - 2876\,n x^{4} - 1310\,n x^{5} - 246\,n x^{6} + 38\,n^{2} + 260\,n^{2} x + 742\,n^{2} x^{2} + 1128\,n^{2} x^{3} + 962\,n^{2} x^{4} + 436\,n^{2} x^{5} + 82\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(98 + 846\,x + 3084\,x^{2} + 6164\,x^{3} + 7306\,x^{4} + 5142\,x^{5} + 1992\,x^{6} + 328\,x^{7} - 51\,n - 434\,n x - 1565\,n x^{2} - 3104\,n x^{3} - 3661\,n x^{4} - 2570\,n x^{5} - 995\,n x^{6} - 164\,n x^{7}\right)\,P_{n-1}'(x) + \left(17 + 173\,x + 759\,x^{2} + 1877\,x^{3} + 2865\,x^{4} + 2767\,x^{5} + 1653\,x^{6} + 559\,x^{7} + 82\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15914
2175849124
3613075112811160
4217144435823924160110768
577363972107434538281539129100064
627532724611190324410029822319345452049929728
7980511293555556115129232462999239680512908032967618638592
834921458840262965285851761746227022658056183141488430408169200180265472
912437318358251201110845714740111538470180908814195047156134787524541718219647049745786880
10442961725654653364477231983384660143314128484922817321112501596976280963676125343673890550032496929480704
111577629284023312319108191133611965368575667483691716621354228096615614978330125731671856732877415215794877531360444164385288192
12561880911026599698978418653735698041965097468750994790232962865923321332823567921342320481359592177782046164987738134352722521788035201598236073984
132001168542516246941612286782484192851010092160149529462657535963587634675610272403611401242113256579111029024539571310968309431163716663883053888609101945937695558512050176
147127267316297906781727313796311245883762850247695197716298774413543957907741867730936624570410315846181787110718022906988895569111193518800749759620761951538835103225245665812510964951646929289216
Technical details
Node id
n-4b9cf0519854296e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b8d211808a3ff6b0

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:      5,       9;
  2:     17,      58,       49;
  3:     61,     307,      511,      281;
  4:    217,    1444,     3582,     3924,      1601;
  5:    773,    6397,    21074,    34538,     28153,      9129;
  6:   2753,   27246,   111903,   244100,    298223,    193454,     52049;
  7:   9805,  112935,   555561,  1512923,   2462999,   2396805,   1290803,    296761;
  8:  34921,  458840,  2629652,  8585176,  17462270,  22658056,  18314148,   8430408,  1692001;
  9: 124373, 1835825, 12011108, 45714740, 111538470, 180908814, 195047156, 134787524, 54171821, 9647049;
%F Formula
The row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (38*n^2-100*n+62+(260*n^2-722*n+462)*x+(742*n^2-2138*n+1396)*x^2+(1128*n^2-3328*n+2200)*x^3+(962*n^2-2876*n+1914)*x^4+(436*n^2-1310*n+874)*x^5+(82*n^2-246*n+164)*x^6)*P(n-1,x) - (51*n-98+(434*n-846)*x+(1565*n-3084)*x^2+(3104*n-6164)*x^3+(3661*n-7306)*x^4+(2570*n-5142)*x^5+(995*n-1992)*x^6+(164*n-328)*x^7)*d/dx P(n-1,x) + (17+173*x+759*x^2+1877*x^3+2865*x^4+2767*x^5+1653*x^6+559*x^7+82*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 9*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 9*x;
P[n_, x_] := P[n, x] = Expand[Together[((38*n^2 - 100*n + 62 + (260*n^2 - 722*n + 462)*x + (742*n^2 - 2138*n + 1396)*x^2 + (1128*n^2 - 3328*n + 2200)*x^3 + (962*n^2 - 2876*n + 1914)*x^4 + (436*n^2 - 1310*n + 874)*x^5 + (82*n^2 - 246*n + 164)*x^6)*P[n - 1, x] - (51*n - 98 + (434*n - 846)*x + (1565*n - 3084)*x^2 + (3104*n - 6164)*x^3 + (3661*n - 7306)*x^4 + (2570*n - 5142)*x^5 + (995*n - 1992)*x^6 + (164*n - 328)*x^7)*D[P[n - 1, x], x] + (17 + 173*x + 759*x^2 + 1877*x^3 + 2865*x^4 + 2767*x^5 + 1653*x^6 + 559*x^7 + 82*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = ((38*n^2 - 100*n + 62 + (260*n^2 - 722*n + 462)*x + (742*n^2 - 2138*n + 1396)*x^2 + (1128*n^2 - 3328*n + 2200)*x^3 + (962*n^2 - 2876*n + 1914)*x^4 + (436*n^2 - 1310*n + 874)*x^5 + (82*n^2 - 246*n + 164)*x^6)*v[i - 1] - (51*n - 98 + (434*n - 846)*x + (1565*n - 3084)*x^2 + (3104*n - 6164)*x^3 + (3661*n - 7306)*x^4 + (2570*n - 5142)*x^5 + (995*n - 1992)*x^6 + (164*n - 328)*x^7)*deriv(v[i - 1], x) + (17 + 173*x + 759*x^2 + 1877*x^3 + 2865*x^4 + 2767*x^5 + 1653*x^6 + 559*x^7 + 82*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)); 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