A028246 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 5\,x + 9\,x^{2} + 7\,x^{3} + 2\,x^{4} - n x - 4\,n x^{2} - 5\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 7\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)

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

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	6													12
3	1	10	29	26												66
4	1	19	101	209	150											480
5	1	36	314	1126	1809	1082										4368
6	1	69	932	5158	13917	18269	9366									47712
7	1	134	2715	21806	87919	190002	210853	94586								608016
8	1	263	7863	88331	499669	1580997	2848477	2737769	1091670							8855040
9	1	520	22784	349642	2668982	11600686	30170960	46597894	39497657	14174522						145083648
10	1	1033	66254	1367464	13725218	78753322	277312388	611858224	826823669	626813813	204495126					2641216512
11	1	2058	193577	5318754	68933490	508276308	2327757354	6880456452	13177070829	15826167490	10851614157	3245265146				52891055616
12	1	4107	568361	20653125	341183064	3171358254	18378890466	69894152106	177881541345	300874085719	325176345573	203522409089	56183135190			1155444326400
13	1	8204	1676310	80248702	1673894975	19338759402	139050095508	661785325020	2149222859679	4800140914960	7268746927766	7140551306214	4110690784225	1053716696762		27344999497728
14	1	16397	4963272	312416362	8172098651	116101477347	1020575596614	5958213118284	24014157650199	67999882425187	135266242247996	185391679982442	166918627365037	88957098135437	21282685940886	696933753434112

Technical details

Node id: n-3fa0e81aa186d9ff · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-179ce8abeeb2cd96

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,     6;
  3: 1,  10,    29,     26;
  4: 1,  19,   101,    209,     150;
  5: 1,  36,   314,   1126,    1809,     1082;
  6: 1,  69,   932,   5158,   13917,    18269,     9366;
  7: 1, 134,  2715,  21806,   87919,   190002,   210853,    94586;
  8: 1, 263,  7863,  88331,  499669,  1580997,  2848477,  2737769,  1091670;
  9: 1, 520, 22784, 349642, 2668982, 11600686, 30170960, 46597894, 39497657, 14174522;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(n-5)*x-(4*n-9)*x^2-(5*n-7)*x^3-(2*n-2)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*x^5)*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[((1 - (n - 5)*x - (4*n - 9)*x^2 - (5*n - 7)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((1 - (n - 5)*x - (4*n - 9)*x^2 - (5*n - 7)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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 Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

Related ?

Same fingerprint ?

A130850 → x ↦ x+1

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741