A056241 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(-4 - 48\,x - 174\,x^{2} - 298\,x^{3} - 270\,x^{4} - 126\,x^{5} - 24\,x^{6} + 3\,n + 56\,n x + 228\,n x^{2} + 414\,n x^{3} + 389\,n x^{4} + 186\,n x^{5} + 36\,n x^{6} + n^{2} - 8\,n^{2} x - 54\,n^{2} x^{2} - 116\,n^{2} x^{3} - 119\,n^{2} x^{4} - 60\,n^{2} x^{5} - 12\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(-2 - 38\,x - 210\,x^{2} - 548\,x^{3} - 782\,x^{4} - 630\,x^{5} - 270\,x^{6} - 48\,x^{7} + 12\,n x + 84\,n x^{2} + 240\,n x^{3} + 360\,n x^{4} + 300\,n x^{5} + 132\,n x^{6} + 24\,n x^{7}\right)\,P_{n-1}'(x) + \left(-4\,x - 32\,x^{2} - 112\,x^{3} - 220\,x^{4} - 260\,x^{5} - 184\,x^{6} - 72\,x^{7} - 12\,x^{8}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	5	5													11
3	1	9	21	14												45
4	1	14	55	82	41											193
5	1	20	115	280	305	122										843
6	1	27	210	731	1278	1095	365									3707
7	1	35	350	1617	3990	5425	3829	1094								16341
8	1	44	546	3192	10347	19688	21874	13124	3281							72097
9	1	54	810	5796	23571	58518	90426	84888	44289	9842						318195
10	1	65	1155	9870	48765	151015	302945	393670	319855	147625	29525					1404491
11	1	77	1595	15972	93621	350471	873257	1467400	1644159	1177297	487157	88574				6199581
12	1	90	2145	24794	169290	748506	2246113	4672278	6748218	6643030	4251537	1594326	265721			27366049
13	1	104	2821	37180	291434	1494922	5280717	13190424	23519418	29766958	26123669	15112864	5181553	797162		120799227
14	1	119	3640	54145	481481	2824822	11541621	33847581	72297141	112707070	126889889	100442377	53011252	16740395	2391485	533233019

Technical details

Node id: n-a25201101abbced1 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-cd2870b9fb32dc0d

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: 1,  2;
  2: 1,  5,   5;
  3: 1,  9,  21,   14;
  4: 1, 14,  55,   82,    41;
  5: 1, 20, 115,  280,   305,   122;
  6: 1, 27, 210,  731,  1278,  1095,   365;
  7: 1, 35, 350, 1617,  3990,  5425,  3829,  1094;
  8: 1, 44, 546, 3192, 10347, 19688, 21874, 13124,  3281;
  9: 1, 54, 810, 5796, 23571, 58518, 90426, 84888, 44289, 9842;

%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) = (n^2+3*n-4-(8*n^2-56*n+48)*x-(54*n^2-228*n+174)*x^2-(116*n^2-414*n+298)*x^3-(119*n^2-389*n+270)*x^4-(60*n^2-186*n+126)*x^5-(12*n^2-36*n+24)*x^6)*P(n-1,x) - (2-(12*n-38)*x-(84*n-210)*x^2-(240*n-548)*x^3-(360*n-782)*x^4-(300*n-630)*x^5-(132*n-270)*x^6-(24*n-48)*x^7)*d/dx P(n-1,x) - (4*x+32*x^2+112*x^3+220*x^4+260*x^5+184*x^6+72*x^7+12*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n - 4 - (8*n^2 - 56*n + 48)*x - (54*n^2 - 228*n + 174)*x^2 - (116*n^2 - 414*n + 298)*x^3 - (119*n^2 - 389*n + 270)*x^4 - (60*n^2 - 186*n + 126)*x^5 - (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] - (2 - (12*n - 38)*x - (84*n - 210)*x^2 - (240*n - 548)*x^3 - (360*n - 782)*x^4 - (300*n - 630)*x^5 - (132*n - 270)*x^6 - (24*n - 48)*x^7)*D[P[n - 1, x], x] - (4*x + 32*x^2 + 112*x^3 + 220*x^4 + 260*x^5 + 184*x^6 + 72*x^7 + 12*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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n - 4 - (8*n^2 - 56*n + 48)*x - (54*n^2 - 228*n + 174)*x^2 - (116*n^2 - 414*n + 298)*x^3 - (119*n^2 - 389*n + 270)*x^4 - (60*n^2 - 186*n + 126)*x^5 - (12*n^2 - 36*n + 24)*x^6)*v[i - 1] - (2 - (12*n - 38)*x - (84*n - 210)*x^2 - (240*n - 548)*x^3 - (360*n - 782)*x^4 - (300*n - 630)*x^5 - (132*n - 270)*x^6 - (24*n - 48)*x^7)*deriv(v[i - 1], x) - (4*x + 32*x^2 + 112*x^3 + 220*x^4 + 260*x^5 + 184*x^6 + 72*x^7 + 12*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A056241 → basis A007318

References

[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