Real-rooted atlas

A113413 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 6\,x + 6\,x^{2} + 2\,x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(42 + 296\,x + 826\,x^{2} + 1192\,x^{3} + 950\,x^{4} + 400\,x^{5} + 70\,x^{6} - 39\,n - 298\,n x - 915\,n x^{2} - 1452\,n x^{3} - 1261\,n x^{4} - 570\,n x^{5} - 105\,n x^{6} + 9\,n^{2} + 66\,n^{2} x + 209\,n^{2} x^{2} + 356\,n^{2} x^{3} + 339\,n^{2} x^{4} + 170\,n^{2} x^{5} + 35\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(18 + 206\,x + 908\,x^{2} + 2072\,x^{3} + 2698\,x^{4} + 2030\,x^{5} + 824\,x^{6} + 140\,x^{7} - 6\,n - 70\,n x - 322\,n x^{2} - 778\,n x^{3} - 1082\,n x^{4} - 874\,n x^{5} - 382\,n x^{6} - 70\,n x^{7}\right)\,P_{n-1}'(x) + \left(1 + 16\,x + 102\,x^{2} + 340\,x^{3} + 660\,x^{4} + 776\,x^{5} + 546\,x^{6} + 212\,x^{7} + 35\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
27384792
317143379317856
4414681926338021377952
5991427798221622283511440773888
623941622946510841221848122864297127686528
7577117791009414702631285427206079317943036547976378880
813933261632818818521846408710139157361851474813801384441441759269376
933638883110265726808300285422347840840214111010816041583610453743529760487550701568
108119238814311697923760424117097022389699124886723342136196087213513663477822390062006350075116845056
111960163535192432157963460945128148217656836864865913118944338029012647080917111317403715795954147135261247747543179264
12473211675836268889942583848761655615079745013964024136331548567130358329591119186311384620769290033251298425959549089118849897441747574784
13114243438855576986330816374370583684462529705515377110689923580305711338380625737526965937390693885998965549706717178456818310908874839614761617674104498749440
1427580711422330217484605252276474019912304775113107852998476728184205151434394156036426141488056602085087590887494171948785801767130285639749326421225609844649041445122008738136960892928
Technical details
Node id
n-4e536b11358e1429 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7a731f1f6f6be712

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:    3,     7;
  2:    7,    38,      47;
  3:   17,   143,     379,     317;
  4:   41,   468,    1926,    3380,     2137;
  5:   99,  1427,    7982,   21622,    28351,    14407;
  6:  239,  4162,   29465,  108412,   218481,   228642,     97127;
  7:  577, 11779,  100941,  470263,  1285427,  2060793,   1794303,    654797;
  8: 1393, 32616,  328188, 1852184,  6408710, 13915736,  18514748,  13801384,   4414417;
  9: 3363, 88831, 1026572, 6808300, 28542234, 78408402, 141110108, 160415836, 104537435, 29760487;
%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) = (9*n^2-39*n+42+(66*n^2-298*n+296)*x+(209*n^2-915*n+826)*x^2+(356*n^2-1452*n+1192)*x^3+(339*n^2-1261*n+950)*x^4+(170*n^2-570*n+400)*x^5+(35*n^2-105*n+70)*x^6)*P(n-1,x) - (6*n-18+(70*n-206)*x+(322*n-908)*x^2+(778*n-2072)*x^3+(1082*n-2698)*x^4+(874*n-2030)*x^5+(382*n-824)*x^6+(70*n-140)*x^7)*d/dx P(n-1,x) + (1+16*x+102*x^2+340*x^3+660*x^4+776*x^5+546*x^6+212*x^7+35*x^8)*d^2/dx^2 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^2 - 39*n + 42 + (66*n^2 - 298*n + 296)*x + (209*n^2 - 915*n + 826)*x^2 + (356*n^2 - 1452*n + 1192)*x^3 + (339*n^2 - 1261*n + 950)*x^4 + (170*n^2 - 570*n + 400)*x^5 + (35*n^2 - 105*n + 70)*x^6)*P[n - 1, x] - (6*n - 18 + (70*n - 206)*x + (322*n - 908)*x^2 + (778*n - 2072)*x^3 + (1082*n - 2698)*x^4 + (874*n - 2030)*x^5 + (382*n - 824)*x^6 + (70*n - 140)*x^7)*D[P[n - 1, x], x] + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*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; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 - 39*n + 42 + (66*n^2 - 298*n + 296)*x + (209*n^2 - 915*n + 826)*x^2 + (356*n^2 - 1452*n + 1192)*x^3 + (339*n^2 - 1261*n + 950)*x^4 + (170*n^2 - 570*n + 400)*x^5 + (35*n^2 - 105*n + 70)*x^6)*v[i - 1] - (6*n - 18 + (70*n - 206)*x + (322*n - 908)*x^2 + (778*n - 2072)*x^3 + (1082*n - 2698)*x^4 + (874*n - 2030)*x^5 + (382*n - 824)*x^6 + (70*n - 140)*x^7)*deriv(v[i - 1], x) + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*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