Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).

A110608 (tail) ↗ nonnegative coefficients

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

Recurrence from A110608

\(\left(12 + 14\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(35 + 25\,x + 46\,n + 35\,n x + 15\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(2 + 11\,x - 13\,x^{2} + n + 7\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6 + 8\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	6	8	1															15
1	20	45	18	1														84
2	70	224	168	32	1													495
3	252	1050	1200	450	50	1												3003
4	924	4752	7425	4400	990	72	1											18564
5	3432	21021	42042	35035	12740	1911	98	1										116280
6	12870	91520	224224	244608	127400	31360	3360	128	1									735471
7	48620	393822	1145664	1559376	1079568	385560	68544	5508	162	1								4686825
8	184756	1679600	5668650	9302400	8139600	3907008	1017450	136800	8550	200	1							30045015
9	705432	7113106	27358100	52762050	56279520	34471206	12166308	2413950	254100	12705	242	1						193536720
10	2704156	29953728	129442896	287650880	364058145	274114368	124366704	33663168	5259870	445280	18216	288	1					1251677700
11	10400600	125550100	602640480	1519156210	2234053250	2010647925	1128784800	395074680	84658860	10689250	743600	25350	338	1				8122425444
12	40116600	524190240	2768379705	7816601520	13136233110	13827613800	9333639315	4063625280	1131350220	196756560	20495475	1192464	34398	392	1			52860229080
13	155117520	2181340125	12574784250	39354417375	74566264500	90225180045	71607285750	37663572375	13100373000	2971843875	427945518	37407825	1847300	45675	450	1		344867425584
14	601080390	9051563520	56572272000	194529216000	410942968800	563578928640	516614017920	320878272000	135370521000	38505392640	7256785536	879610368	65447200	2777600	59520	512	1	2254848913647

Technical details

Node id: n-f46f66d0d087b029 · revision 0 · JSON
Definition source: oeis_catalog:A110608
Fingerprint: f6-98b5f32915261152

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

Row sums ? A005809

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      6,       8,        1;
  1:     20,      45,       18,        1;
  2:     70,     224,      168,       32,        1;
  3:    252,    1050,     1200,      450,       50,        1;
  4:    924,    4752,     7425,     4400,      990,       72,        1;
  5:   3432,   21021,    42042,    35035,    12740,     1911,       98,       1;
  6:  12870,   91520,   224224,   244608,   127400,    31360,     3360,     128,      1;
  7:  48620,  393822,  1145664,  1559376,  1079568,   385560,    68544,    5508,    162,     1;
  8: 184756, 1679600,  5668650,  9302400,  8139600,  3907008,  1017450,  136800,   8550,   200,   1;
  9: 705432, 7113106, 27358100, 52762050, 56279520, 34471206, 12166308, 2413950, 254100, 12705, 242, 1;

%F Formula

The row polynomials satisfy (4*n^2+14*n+12)*P(n,x) = (15*n^2+46*n+35+(12*n^2+35*n+25)*x)*P(n-1,x) + (n+2+(7*n+11)*x-(8*n+13)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 8*x + x^2.

%t Mathematica

P[0, x_] := 6 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n^2 + 46*n + 35 + (12*n^2 + 35*n + 25)*x)*P[n - 1, x] + (n + 2 + (7*n + 11)*x - (8*n + 13)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^2 + 46*n + 35 + (12*n^2 + 35*n + 25)*x)*v[i - 1] + (n + 2 + (7*n + 11)*x - (8*n + 13)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12)); 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 (−∞, 0)
by Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Used in 6

Related ?

Same fingerprint ?

A110608 → reverse

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.