A026641 -id:A026641 - cp's OEIS frontend

A026647 a(n) = Sum_{k=0..floor(n/2)} A026637(n-k, k).

1, 1, 2, 3, 6, 10, 17, 27, 45, 73, 119, 192, 312, 505, 818, 1323, 2142, 3466, 5609, 9075, 14685, 23761, 38447, 62208, 100656, 162865, 263522, 426387, 689910, 1116298, 1806209, 2922507, 4728717, 7651225, 12379943, 20031168

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,-1,-1).

Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026643.
Cf. A026644, A026645, A026646, A026966, A026967, A026968, A026969.
Cf. A026970.

[1] cat [n le 5 select Binomial(n, Floor(n/2)) else Self(n-2) +Self(n-3) +2*Self(n-4) +Self(n-5) +3: n in [1..40]]; // G. C. Greubel, Jul 01 2024

a[n_]:= a[n]= If[n<6, Binomial[n, Floor[n/2]], a[n-2] +a[n-3] +2*a[n- 4] +a[n-5] +3]; (* a = A026647 *)
Table[a[n], {n,0,40}] (* G. C. Greubel, Jul 01 2024 *)
LinearRecurrence[{1,1,0,1,-1,-1},{1,1,2,3,6,10,17},40] (* Harvey P. Dale, Jan 19 2025 *)

@CachedFunction
def a(n): # a = A026647
    if n<6: return binomial(n, n//2)
    else: return a(n-2) + a(n-3) + 2*a(n-4) + a(n-5) + 3
[a(n) for n in range(41)] # G. C. Greubel, Jul 01 2024

G.f.: (1 + x^5 + x^6)/((1-x^4)*(1-x-x^2)).
From G. C. Greubel, Jul 01 2024: (Start)
a(n) = [n=0] - (3/4) + (1/4)*(-1)^n - (1/10)*2^((1-(-1)^n)/2)*(-1)^floor((n+1)/2) + (3/5)*LucasL(n+1).
a(n) = (1/20)*( 12*LucasL(n+1) + 5*(-1)^n - 15 - 2*cos(n*Pi/2) + 4*sin(n*Pi/2) ) + [n=0].
a(n) = a(n-2) + a(n-3) + 2*a(n-4) + a(n-5) + 3, with a(0) = a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 6, a(5) = 10. (End)

A127361 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*(-2)^(n-k).

Original entry on oeis.org

1, -1, 4, -7, 22, -46, 130, -295, 790, -1870, 4864, -11782, 30148, -73984, 187534, -463687, 1168870, -2902870, 7293640, -18161170, 45541492, -113576596, 284470564, -710118262, 1777323772, -4439253196, 11105933440, -27749232700, 69403169200

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform is 3^n. In general, for r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*(-r)^(n-k) has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+2*x) under the Chebyshev mapping g(x) -> (1/sqrt(1-4*x^2)) * g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
Second binomial transform is A026641. - Philippe Deléham, Mar 14 2007
Signed version of A100098. - Philippe Deléham, Nov 25 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A026641, A061554.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1+2*x-Sqrt(1-4*x^2))/(2*Sqrt(1-4*x^2)*(1+x-Sqrt(1-4*x^2))) )); // G. C. Greubel, Feb 17 2019

a:=n->add(binomial(n,floor(k/2))*(-2)^(n-k),k=0..n): seq(a(n),n=0..30); # Muniru A Asiru, Feb 18 2019

CoefficientList[Series[(1/Sqrt[1-4*x^2])*(1+x*(1-Sqrt[1-4*x^2]) / (2*x^2)) /(1+2*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

my(x='x+O('x^30)); Vec( (1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ) \\ G. C. Greubel, Feb 17 2019

((1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 17 2019

G.f.: (1/sqrt(1-4*x^2))(1+x*c(x^2))/(1+2*x*c(x^2)), with c(x) = (1 - sqrt(1-4*x))/(2*x).
a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - Philippe Deléham, Nov 25 2007
a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - Philippe Deléham, Dec 04 2009
Conjecture: 2*n*a(n) + (5*n-4)*a(n-1) - 2*(4*n-3)*a(n-2) - 20*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
a(n) ~ (-1)^n * 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 13 2014

More terms from Vincenzo Librandi, Feb 15 2014

A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5n-2k-1,n-2*k).

Original entry on oeis.org

1, 4, 37, 376, 4013, 44064, 492871, 5585080, 63901421, 736575316, 8540549322, 99503540008, 1163910870767, 13660217796736, 160782910480936, 1897131524755896, 22433316399634669, 265775992115557076, 3154067508987675679, 37487016824453703920, 446148092364247390618

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A026641, A147855, A183160.

Cf. A001449, A079589, A079678, A385632, A386812.

A371753 := proc(n)
    add( binomial(5*n-2*k-1,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A371753(n),n=0..50) ; # R. J. Mathar, Sep 27 2024

a(n) = sum(k=0, n\2, binomial(5*n-2*k-1, n-2*k));

a(n) = [x^n] 1/((1-x^2) * (1-x)^(4*n)).
a(n) ~ 5^(5*n + 3/2) / (3 * sqrt(Pi*n) * 2^(8*n + 5/2)). - Vaclav Kotesovec, Apr 05 2024
Conjecture D-finite with recurrence +1024*n*(796184150374453*n -1374782084855770) *(4*n-3)*(2*n-1)*(4*n-1)*a(n) +64*(-4720591427354845074*n^5 +16046598674673412696*n^4 -14164434258362644374*n^3 -6132680339747354209*n^2 +16406971563067867560*n -7312237120275595200)*a(n-1) +40*(-4968388566264801507*n^5 +51044954667717039608*n^4 -218029351288077225930*n^3 +471970442274586326109*n^2 -511707487331990011785*n +221366817798624198360)*a(n-2) -25*(5*n-11) *(719005061479699*n -1438086256867727)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Sep 27 2024
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
G.f.: g^2/((-1+2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. (End)
G.f.: B(x)^2/(1 + 6*(B(x)-1)/5), where B(x) is the g.f. of A001449. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-5+9*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 16 2025

A386957 a(n) = Sum_{k=0..n} 8^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 11, 114, 1163, 11806, 119646, 1211820, 12271179, 124251318, 1258065866, 12737997724, 128972535582, 1305848105836, 13221716621852, 133869898347264, 1355432788629963, 13723757247851046, 138953043155444562, 1406899565919247884, 14244858120395937738, 144229188529316725956

Offset: 0

Seiichi Manyama, Aug 11 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A000302, A000984, A026641, A141223, A377011.

Cf. A386958, A386960.

[&+[8^k * Binomial(2*n+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 14 2025

Table[Sum[8^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 14 2025 *)

a(n) = sum(k=0, n, 8^k*binomial(2*n+1, n-k));

a(n) = [x^n] 1/((1-9*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k,n-k).
G.f.: 2/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).
D-finite with recurrence 8*n*a(n) +(-113*n+16)*a(n-1) +162*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025

A101850 A Catalan transform of Pell(n+1).

Original entry on oeis.org

1, 2, 7, 26, 100, 392, 1555, 6218, 25006, 100988, 409162, 1661948, 6764194, 27575732, 112570675, 460058906, 1881978694, 7704907724, 31566153058, 129400608044, 530734613920, 2177792579072, 8939838222718, 36711025334948

Offset: 0

Paul Barry, Dec 18 2004

A Catalan transform of the Pell numbers A000129(n+1) under the mapping G(x) -> G(x*c(x)), c(x) the g.f. of A000108. The inverse mapping is H(x) -> H(x*(1-x)).
Hankel transform is 3^n. - Paul Barry, Jan 19 2009
Row sums of the Riordan matrix (1/(x+sqrt(1-4*x)),(1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x)))) (A188513). - Emanuele Munarini, Apr 02 2011
Equals the INVERT transform of A026641: (1, 1, 4, 13, 46, 166, ...). Example: a(4) = 100 = (1, 1, 2, 7, 26) dot (46, 13, 4, 1, 1) = (46 + 13 + 8 + 7 + 26 ) = 100. - Gary W. Adamson, Jan 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012

Cf. A081696, A026641, A188513.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2/(3*Sqrt(1-4*x)+2*x-1) )); // G. C. Greubel, Feb 18 2019

CoefficientList[Series[(1-2x+3Sqrt[1-4x])/(4-16x-2x^2),{x,0,24}],x] (* Emanuele Munarini, Apr 02 2011 *)
CoefficientList[Series[2/(3*Sqrt[1-4*x]+2*x-1), {x,0,30}], x] (* or *) RecurrenceTable[{2*(n+3)*a[n+3] -4*(4*n+9)*a[n+2]+(31*n+45)*a[n+1] + 2*(2*n+3)*a[n]==0, a[0]==1, a[1]==2, a[2]==7}, a, {n, 0, 30}](* G. C. Greubel, Feb 18 2019 *)

makelist(sum(binomial(2*n+1,n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2),k,0,n),n,0,12); /* Emanuele Munarini, Apr 02 2011 */

my(x='x+O('x^30)); Vec(2/(3*sqrt(1-4*x)+2*x-1)) \\ G. C. Greubel, Feb 18 2019

(2/(3*sqrt(1-4*x)+2*x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

G.f.: 2/(3*sqrt(1-4*x) + 2*x - 1).
a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*A000129(k+1).
a(n) = Sum_{k=0..n} A039599(n,k)*A016116(k). - Philippe Deléham, Oct 29 2008
G.f.: 1/(1-2*x-3*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-... (continued fraction). - Paul Barry, Jan 19 2009
From Emanuele Munarini, Apr 02 2011: (Start)
a(n) = [x^n] (1-2*x)/((1-2*x-x^2)(1-x)^(n+1)).
a(n) = Sum_{k=0..n} binomial(2*n+1, n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2).
Recurrence: 2*(n+3)*a(n+3) -4*(4*n+9)*a(n+2) +(31*n+45)*a(n+1) +2*(2*n+3)*a(n) = 0. (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:
  2, 3, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)
a(n) ~ (1/4)*(2 + 3/sqrt(2))^n. - Vaclav Kotesovec, Oct 17 2012

A158815 Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 13, 5, 1, 1, 46, 16, 6, 1, 1, 166, 58, 19, 7, 1, 1, 610, 211, 71, 22, 8, 1, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1, 32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 27 2009

The left factor of the matrix product is the triangle which starts
   1;
   2,  1;
   6,  3, 1;
  20, 10, 4, 1;
a row-reversed version of A046899, equivalent to the triangular view of the array A092392. The right factor is the inverse of the matrix A007318, which is A130595.
Swapping the two factors, A007318^(-1) * A046899(row-reversed) would generate A158793.
Riordan array (f(x), g(x)) where f(x) is the g.f. of A026641 and where g(x) is the g.f. of A000957. - Philippe Deléham, Dec 05 2009
T(n,k) is the number of nonnegative paths consisting of upsteps U=(1,1) and downsteps D=(1,-1) of length 2n with k low peaks. (A low peak has its peak vertex at height 1.) Example: T(3,1)=5 counts UDUUUU, UDUUUD, UDUUDU, UDUUDD, UUDDUD. - David Callan, Nov 21 2011
Matrix product P^2 * Q * P^(-2), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A171243. - Peter Bala, Jul 13 2021

			The triangle starts
       1;
       1,     1;
       4,     1,     1;
      13,     5,     1,    1;
      46,    16,     6,    1,    1;
     166,    58,    19,    7,    1,   1;
     610,   211,    71,   22,    8,   1,   1;
    2269,   781,   261,   85,   25,   9,   1,  1;
    8518,  2620,   976,  316,  100,  28,  10,  1,  1;
   32206, 11006,  3676, 1196,  376, 116,  31, 11,  1, 1;
  122464, 41746, 13938, 4544, 1442, 441, 133, 34, 12, 1, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Filippo Disanto, Andrea Frosini and Simone Rinaldi, Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.

Cf. A000984, A026641, A046899, A158793, A171243.

A158815 := proc (n, k)
  add((-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k), j = 0..n);
end proc:
seq(seq(A158815(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T[n_,k_]:= T[n,k]= Sum[(-1)^(j+k)*Binomial[j,k]*Binomial[2*n-j,n], {j,0,n}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)

def A158815(n,k): return sum( (-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k) for j in (0..n) )
flatten([[A158815(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

Sum_{k=0..n} T(n,k) = A046899(n).
T(n,0) = A026641(n).
Sum_{k=0..n} T(n,k)*x^k = A026641(n), A000984(n), A001700(n), A000302(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Dec 03 2009
T(n, k) = Sum_{j=0..n} binomial(j, k)*binomial(2*n-j, n). - Peter Bala, Jul 13 2021

A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

Original entry on oeis.org

1, 2, 6, 19, 65, 231, 841, 3110, 11628, 43834, 166298, 634140, 2428336, 9331688, 35967462, 138987715, 538287881, 2088842463, 8119916647, 31613327405, 123251518641, 481125828853, 1880262896537, 7355767408395, 28803717914791, 112887697489907, 442784607413427

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1664

Partial sums of A026641. - Seiichi Manyama, Jan 30 2023
Cf. A000108, A006134, A079309, A105872, A120305, A306409.
Cf. A191993, A360186, A360212.

Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(i!*j!), {i, 0, j}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 04 2019 *)

a(n) = sum(i=0, n, sum(j=i, n, (-1)^(i+j)*(i+j)!/(i!*j!)));

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 29 2023

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^3*(2/(1+sqrt(1-4*x)))^3))) \\ Seiichi Manyama, Jan 29 2023

a(n) = (A006134(n) + A120305(n))/2.
From Vaclav Kotesovec, Apr 04 2019: (Start)
Recurrence: 2*n*a(n) = (9*n-4)*a(n-1) - (3*n-2)*a(n-2) - 2*(2*n-1)*a(n-3).
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). (End)
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)^3) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

A371798 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-2k-1,n-2*k).

Original entry on oeis.org

1, 1, 2, 7, 26, 96, 356, 1331, 5014, 19006, 72412, 277058, 1063856, 4097510, 15823432, 61245987, 237536326, 922906150, 3591500972, 13996328322, 54614894396, 213360770840, 834409399672, 3266370155262, 12797894251276, 50184309630196, 196936674150296

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Sergi Elizalde, Nadia Lafrenière, Joel Brewster Lewis, Erin McNicholas, Jessica Striker, and Amanda Welch, Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks, arXiv:2412.16368 [math.CO], 2024. See p. 13.

Cf. A026641, A120305.

Table[Sum[(-1)^k Binomial[2n-2k-1,n-2k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Oct 31 2024 *)

a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k-1, n-2*k));

a(n) = [x^n] 1/((1+x^2) * (1-x)^n).
a(n) = binomial(2*n-1, n)*hypergeom([1, (1-n)/2, -n/2], [1/2-n, 1-n], -1). - Stefano Spezia, Apr 06 2024
a(n) ~ 2^(2*n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 07 2024
Conjectured g.f.: 1 + x*(4 - 10*x + 8*x^2)/(2 - 11*x + 14*x^2 - 8*x^3 + (2 - 3*x)*sqrt(1 - 4*x)) (see Elizalde et al. at p. 13). - Stefano Spezia, Dec 27 2024

A191354 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,2), and (2,1).

Original entry on oeis.org

1, 1, 3, 9, 25, 75, 227, 693, 2139, 6645, 20757, 65139, 205189, 648427, 2054775, 6526841, 20775357, 66251247, 211617131, 676930325, 2168252571, 6953348149, 22322825865, 71735559255, 230735316795, 742773456825, 2392949225565, 7714727440755, 24888317247705, 80341227688095

Offset: 0

Joerg Arndt, Jun 30 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-2*x-3*x^2-4*x^3) )); // G. C. Greubel, Feb 18 2019

a[n_]:= Sum[Binomial[2k, k]*Sum[Binomial[j, n-k-j]*Binomial[k, j]*2^(j-k) *3^(-n+k+2j)*4^(n-k-2j), {j, 0, k}], {k, 0, n}];
Array[a, 30, 0] (* Jean-François Alcover, Jul 21 2018, after Vladimir Kruchinin *)
CoefficientList[Series[1/Sqrt[1-2*x-3*x^2-4*x^3], {x, 0, 30}], x] (* G. C. Greubel, Feb 18 2019 *)

a(n):=sum(binomial(2*k,k) * sum(binomial(j,n-k-j) * 2^(j-k) * binomial(k,j) * 3^(-n+k+2*j) * 4^(n-k-2*j),j,0,k),k,0,n); /* Vladimir Kruchinin, Feb 27 2016 */

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,2], [2,1]];
/* Joerg Arndt, Jun 30 2011 */

my(x='x+O('x^30)); Vec(1/sqrt(1-2*x-3*x^2-4*x^3)) \\ G. C. Greubel, Feb 18 2019

(1/sqrt(1-2*x-3*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

G.f.: 1/sqrt(1-2*x-3*x^2-4*x^3). - Mark van Hoeij, Apr 16 2013
G.f.: Q(0), where Q(k) = 1 + x*(2+3*x+4*x^2)*(4*k+1)/( 4*k+2 - x*(2+3*x+4*x^2)*(4*k+2)*(4*k+3)/(x*(2+3*x+4*x^2)*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013
a(n) = Sum_{k=0..n} (binomial(2*k,k) * Sum_{j=0..k} (binomial(j,n-k-j) *binomial(k,j)*2^(j-k)*3^(-n+k+2*j)*4^(n-k-2*j))). - Vladimir Kruchinin, Feb 27 2016
D-finite with recurrence: +(n)*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020

A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 6, 34, 188, 1026, 5556, 29940, 160824, 862018, 4613636, 24667644, 131795912, 703812916, 3757135752, 20051429544, 106992663408, 570827898306, 3045193326372, 16244056119084, 86646747723048, 462161936699196, 2465043081687192, 13147597801986264, 70123266087502608

Offset: 0

Seiichi Manyama, Aug 11 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A293490, A383832, A386942.

Cf. A000302, A000984, A026641, A141223, A386957.

[&+[3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025

Table[Sum[3^k * Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)

a(n) = sum(k=0, n, 3^k*binomial(2*n+1, n-k));

a(n) = [x^n] 1/((1-4*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 4^k * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1) ).
a(n) ~ 2^(4*n+2) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*a(n) +2*(-14*n+3)*a(n-1) +32*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025

cp's OEIS Frontend

A026647 a(n) = Sum_{k=0..floor(n/2)} A026637(n-k, k).

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A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5*n-2*k-1,n-2*k).

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A386957 a(n) = Sum_{k=0..n} 8^k * binomial(2*n+1,n-k).

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A101850 A Catalan transform of Pell(n+1).

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A158815 Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.

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A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

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A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5n-2k-1,n-2*k).

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