A104507 -id:A104507 - cp's OEIS frontend

A246437 Expansion of (1/2)(1/(x+1)+1/(sqrt(-3x^2-2*x+1))).

1, 0, 2, 3, 10, 25, 71, 196, 554, 1569, 4477, 12826, 36895, 106470, 308114, 893803, 2598314, 7567465, 22076405, 64498426, 188689685, 552675364, 1620567764, 4756614061, 13974168191, 41088418150, 120906613076, 356035078101, 1049120176954

Offset: 0

Vladimir Kruchinin, Nov 14 2014

a(2101) has 1001 decimal digits. - Michael De Vlieger, Apr 25 2016
This is the analog for Coxeter type B of Motzkin sums (A005043) for Coxeter type A, see the article by Athanasiadis and Savvidou. - F. Chapoton, Jul 20 2017
Number of compositions of n into exactly n nonnegative parts avoiding part 1. a(4) = 10: 0004, 0022, 0040, 0202, 0220, 0400, 2002, 2020, 2200, 4000. - Alois P. Heinz, Aug 19 2018
From Peter Bala, Jan 07 2022: (Start)
The binary transform is A088218. The inverse binomial transform is a signed version of A027306 and the second inverse binomial transform is a signed version of A027914.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..2100
Christos A. Athanasiadis and Christina Savvidou, The Local h-Vector of the Cluster Subdivision of a Simplex, Séminaire Lotharingien de Combinatoire 66 (2012), Article B66c.
Eric Marberg, On some actions of the 0-Hecke monoids of affine symmetric groups, arXiv:1709.07996 [math.CO], 2017.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 24.

Cf. A002426, A005043, A027306, A027914, A088218, A104507, A370616, A386548.

CoefficientList[Series[(1/2) (1 / (x + 1) + 1 / (Sqrt[-3 x^2 - 2 x + 1])), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
Table[(-1)^n (Hypergeometric2F1[1/2, -n, 1, 4] + 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)
Table[Sum[Binomial[n, k] Binomial[n - k - 1, n - 2 k], {k, 0, n/2}], {n, 0, 28}] (* Michael De Vlieger, Apr 25 2016 *)

a(n):=sum(binomial(n,k)*binomial(n-k-1,n-2*k),k,0,n/2);

def a(n):
    if n < 3: return [1,0,2][n]
    return n*hypergeometric([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4)
[simplify(a(n)) for n in (0..28)] # Peter Luschny, Nov 14 2014

a(n) = Sum_{k = 0..n/2} binomial(n,k)*binomial(n-k-1,n-2*k).
A(x) = 1 + x*B'(x)/B(x), where B(x) = (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)) is the o.g.f. of A005043.
a(n) = n*hypergeom([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4) for n>=3. - Peter Luschny, Nov 14 2014
a(n) ~ 3^(n+1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 14 2014
a(n) = (-1)^n*(hypergeom([1/2, -n], [1], 4) + 1)/2. - Vladimir Reshetnikov, Apr 25 2016
D-finite with recurrence: n*(a(n) - a(n-1)) = (5*n-6)*a(n-2) + 3*(n-2)*a(n-3). - Vladimir Reshetnikov, Oct 13 2016
a(n) = [x^n]( (1 - x + x^2)/(1 - x) )^n. - Peter Bala, Jan 07 2022

A104505 Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0.

Original entry on oeis.org

1, 1, -1, -1, -2, 1, -5, 0, 3, -1, -5, 8, 2, -4, 1, 11, 15, -10, -5, 5, -1, 41, -6, -30, 10, 9, -6, 1, 29, -77, -14, 49, -7, -14, 7, -1, -125, -120, 112, 56, -70, 0, 20, -8, 1, -365, 117, 288, -126, -126, 90, 12, -27, 9, -1, -131, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1, 1409, 946, -1265, -495, 858, 33, -363, 110, 55, -44

Offset: 0

Paul D. Hanna, Mar 11 2005

Matrix inverse is triangle A104509 and is related to Fibonacci numbers. Column 0 equals A098331, with g.f.: 1/sqrt(1-2*x+5*x^2). Column 1 equals A104506, with g.f.: ((1-x)/sqrt(1-2*x+5*x^2)-1)/(2*x). Row sums equal A104507. Absolute row sums equal A104508.

Array (1/sqrt(1-2x+5x^2), (1-x-sqrt(1-2x+5x^2))/(2x)), in Riordan array notation. Product of A120616 by A007318. - Paul Barry, Jun 17 2006

			Rows begin:
1;
1,-1;
-1,-2,1;
-5,0,3,-1;
-5,8,2,-4,1;
11,15,-10,-5,5,-1;
41,-6,-30,10,9,-6,1;
29,-77,-14,49,-7,-14,7,-1;
-125,-120,112,56,-70,0,20,-8,1;
-365,117,288,-126,-126,90,12,-27,9,-1;
-131,770,45,-540,90,228,-105,-30,35,-10,1; ...

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Cf. A104509, A098331, A007440, A104506, A104507, A104508, A108044.

T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)

PARI
```
T(n,k)=if(n
				
```

T(n, 0) = A098331(n). T(n, 1) = n*A007440(n) (n>0).
Column k has e.g.f. exp(x)*Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k. - Paul Barry, Jun 17 2006
From Peter Bala, Jun 29 2015: (Start)
Matrix factorization in the Riordan group: ( 1/(1 - x), x/(1 - x) ) * ( 1/sqrt(1 + 4*x^2), (1 - sqrt(1 + 4*x^2))/(2*x) ) = A007318 * signed version of A108044.
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - x - sqrt(1 - 2*x + 5*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = x^2 + x - 1. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A370616 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^n.

Original entry on oeis.org

1, 0, 2, 3, 14, 35, 125, 371, 1238, 3909, 12847, 41580, 136577, 447187, 1473341, 4855703, 16053830, 53138243, 176233967, 585202261, 1945964079, 6478043120, 21588979876, 72016891508, 240452892569, 803489258285, 2686964354375, 8991840800136, 30110638705889

Offset: 0

Seiichi Manyama, Apr 30 2024

nonn

Cf. A046736, A213684, A104507, A246437, A386548.

Table[Sum[Binomial[-1 - k + n, -2*k + n] Binomial[-1 + k + n, k], {k, 0, n/2}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 30 2025 *)

a(n, s=2, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));

a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(n-k-1,n-2*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x-x^2) / (1-x) ).
From Peter Bala, 26 Jul 2025: (Start)
a(n) = n * hypergeom([1 + n, 1 - n/2, 3/2 - n/2], [2, 2 - n], -4) for n >= 3.
P-recursive: 5*n*(74*n^3-493*n^2+1075*n-766)*(n-1)*a(n) = 2*(n-1)*(296*n^4-2120*n^3+5393*n^2-5716*n+2100)*a(n-1) + 2*(1184*n^5-10256*n^4+34088*n^3-53995*n^2+40397*n-11250)*a(n-2) - 2*(n-3)*(2*n-5)*(74*n^3-271*n^2+311*n-110)*a(n-3) with a(0) = 1, a(1) = 0 and a(2) = 2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k. (End)
a(n) ~ sqrt(1/12 + sqrt(10/37)*(sin(arcsin((13*sqrt(37/10))/40)/3)/3)) * (8*((1 + sqrt(34)*cos(arccos(2461/(1088*sqrt(34)))/3))/15))^n / sqrt(Pi*n). - Vaclav Kotesovec, Jul 30 2025

A386548 a(n) = [x^n] ((1 - x)/(1 - x + x^2))^n.

Original entry on oeis.org

1, 0, -2, -3, 6, 25, 1, -147, -218, 591, 2223, -484, -14871, -18759, 68353, 222697, -116058, -1629671, -1656989, 8275203, 23266031, -20154144, -184550412, -141418628, 1019061001, 2468408775, -3122976521, -21213927840, -10837119735, 126256071125, 262294667301, -456407675223

Offset: 0

Peter Bala, Jul 25 2025

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.

Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 5 and all positive integers n and k.

Cf. A104507, A246437, A364374, A370616.

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 0 elif n = 2 then -2 else
( 2*(n-1)*(2*n-3)*(19*n^2-60*n+36)*a(n-1) - 2*(190*n^4-1170*n^3+2519*n^2-2229*n+666)*a(n-2) - 2*(n-3)*(2*n-3)*(19*n^2-41*n+18)*a(n-3) )/(3*n*(n-1)*(19*n^2-79*n+78)) fi; end:
seq(a(n), n = 0..30);

a[n_]:=SeriesCoefficient[((1 - x)/(1 - x + x^2))^n,{x,0,n}]; Array[a,32,0] (* Stefano Spezia, Jul 29 2025 *)

a(n) = my(x='x+O('x^(n+1))); polcoef(((1 - x)/(1 - x + x^2))^n, n); \\ Michel Marcus, Aug 03 2025

a(n) = Sum_{k = 0..floor(n/2)} binomial(-n, k)*binomial(n-k-1, n-2*k) = Sum_{k = 0
..floor(n/2)} (-1)^k*binomial(n+k-1, k)*binomial(n-k-1, n-2*k). Cf. A246437.
a(n) = -n*hypergeom([n+1, 1 - (1/2)*n, 3/2 - (1/2)*n], [2, 2 - n], 4) for n >= 3.
P-recursive: 3*n*(n - 1)*(19*n^2 - 79*n + 78)*a(n) = 2*(n - 1)*(2*n - 3)*(19*n^2 - 60*n + 36)*a(n-1) - 2*(190*n^4 - 1170*n^3 + 2519*n^2 - 2229*n + 666)*a(n-2) - 2*(n - 3)*(2*n - 3)*(19*n^2 - 41*n + 18)*a(n-3) with a(0) = 1, a(1) = 0 and a(2) = -2.
exp( Sum_{n >= 1} a(n)*(-x)^n/n ) = 1 - x^2 + x^3 + 2*x^4 - 6*x^5 - x^6 +  ... is the g.f. of A364374.

A104506 Column 1 of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1 + x - x^2)^n.

Original entry on oeis.org

0, -1, -2, 0, 8, 15, -6, -77, -120, 117, 770, 946, -1728, -7735, -6930, 22800, 76960, 42245, -282150, -751640, -125800, 3341205, 7145710, -2002725, -38228232, -65418925, 55550014, 424605078, 566938400, -936604097, -4587287310

Offset: 0

Paul D. Hanna, Mar 11 2005

sign

P. Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq. 14 (2011) # 11.4.3, page 8.

Cf. A104505, A007440, A104507, A104508, A104509.

{a(n)=polcoeff(((1-x)/sqrt(1-2*x+5*x^2+x^2*O(x^n))-1)/(2*x),n)}

G.f.: ((1-x)/sqrt(1-2*x+5*x^2) - 1)/(2*x).
a(n) = (-1)^n*n*A007440(n) (reversion of g.f. for Fibonacci numbers).
a(n) = -Sum_{k=0..floor(n/2)} C(n, k)*C(n-k, k+1)*(-1)^k. - Paul Barry, May 02 2005
E.g.f.: -exp(x)Bessel_I(1,2*i*x)/i, i=sqrt(-1). - Paul Barry, Feb 10 2006
-(n-1)*(n+1)*a(n) + n*(2*n-1)*a(n-1) - 5*n*(n-1)*a(n-2) = 0. - R. J. Mathar, Aug 17 2017

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A246437 Expansion of (1/2)*(1/(x+1)+1/(sqrt(-3*x^2-2*x+1))).

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A104505 Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0.

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A370616 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^n.

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A386548 a(n) = [x^n] ((1 - x)/(1 - x + x^2))^n.

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A104506 Column 1 of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1 + x - x^2)^n.

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A246437 Expansion of (1/2)(1/(x+1)+1/(sqrt(-3x^2-2*x+1))).