A081671 -id:A081671 - cp's OEIS frontend

A246876 G.f.: 1 / AGM(1-12x, sqrt((1-4x)(1-36x))).

1, 16, 324, 7744, 206116, 5875776, 175191696, 5386385664, 169300977444, 5410164352576, 175128910042384, 5727842622630144, 188931648862083856, 6276176070222305536, 209747841324097564224, 7046053064278540084224, 237764385841359952067364, 8054915184317632144620096

Offset: 0

Paul D. Hanna, Sep 06 2014

nonn

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by
1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*[Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k)]^2,
and consists of integer coefficients when 4|(q-p).
Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

			G.f.: A(x) = 1 + 16*x + 324*x^2 + 7744*x^3 + 206116*x^4 + 5875776*x^5 +...
where the square-root of the terms yields A081671:
[1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, ...]
the g.f. of which is 1/sqrt((1-2*x)*(1-6*x)).

Seiichi Manyama, Table of n, a(n) for n = 0..644

Cf. A081671, A168597, A246467.

{a(n)=polcoeff( 1 / agm(1-12*x, sqrt((1-4*x)*(1-36*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=sum(k=0,n,2^(n-k)*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))

a(n) = A081671(n)^2 = [Sum_{k=0..n} 2^(n-k) * C(n,k) * C(2*k,k)]^2.
G.f.:  1 / AGM((1-2*x)*(1+6*x), (1+2*x)*(1-6*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) ~ 2^(2*n - 1) * 3^(2*n + 1) / (Pi*n). - Vaclav Kotesovec, Dec 10 2018

A329073 a(n) = (1/n)Sum_{k=0..n-1} (40k+13)(-1)^k50^(n-1-k)T_k(4,1)T_k(1,-1)^2, where T_k(b,c) denotes the coefficient of x^k in the expansion of (x^2+bx+c)^k.

Original entry on oeis.org

13, 219, 7858, 221525, 9253710, 375158958, 16882409364, 736344816813, 32964312771550, 1471835619627770, 66910145732699964, 3061043035494001682, 141458526138008430124, 6567714993530314856700, 306628434270114823521000, 14370411994543866356077725, 676259546148988495771751550

Offset: 1

Zhi-Wei Sun, Nov 03 2019

nonn

Conjecture 1: (i) a(n) is a positive integer for each n > 0; also, a(n) is odd if and only if n is a power of two. Moreover, we have the identity Sum_{k>=0} ((40k+13)/(-50)^k)*T_k(4,1)*T_k(1,-1)^2 = 55*sqrt(15)/(9*Pi).
(ii) Let p > 5 be a prime. Then Sum_{k=0..p-1} ((40k+13)/(-50)^k)*T_k(4,1)* T_k(1,-1)^2 == (p/3)*(12 + 5*Leg(3/p) + 22*Leg(p/15)) (mod p^2), where Leg(a/p) denotes the Legendre symbol. Also, for the sum S(p) = Sum_{k=0..p-1} T_k(4,1)* T_k(1,-1)^2/(-50)^k, if Leg(-5/p) = -1 then S(p) == 0 (mod p^2); if p == 1,9 (mod 20) and p = x^2 + 5*y^2 with x and y integers then S(p) == 4x^2-2p (mod p^2); if p == 3,7 (mod 20) and 2p = x^2 + 5*y^2 with x and y integers then S(p) == 2x^2-2p (mod p^2).
Conjecture 2: (i) For any n > 0, the number b(n):=(1/n)*Sum_{k=0..n-1} (40k+27)*(-6)^(n-1-k)*T_k(4,1)*T_k(1,-1)^2 is an integer. Moreover, b(n) is odd if and only if n is a power of two.
(ii) Let p > 3 be a prime. Then Sum_{k=0..p-1} ((40k+27)/(-6)^k)*T_k(4,1)* T_k(1,-1)^2 == (p/9)*(55*Leg(-5/p) + 198*Leg(3/p)-10) (mod p^2). Also, for the sum T(p) = Sum_{k=0..p-1} T_k(4,1)*T_k(1,-1)^2/(-6)^k, if Leg(-5/p) = -1 then T(p) == 0 (mod p^2); if p == 1,9 (mod 20) and p = x^2 + 5*y^2 with x and y integers then T(p) == Leg(p/3)*(4x^2-2p) (mod p^2); if p == 3,7 (mod 20) and 2p = x^2 + 5*y^2 with x and y integers then T(p) == Leg(p/3)(2p-2x^2) (mod p^2).

			a(1) = 13 since (40*0+13)*(-1)^0*50^(1-1-0)*T_0(4,1)*T_0(1,-1)^2/1 = 13/1 = 13.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. in Math. & Stat., Vol. 101, Springer, New York, 2014, pp. 257-312. Also available from arXiv:1101.0600 [math.NT], 2011-2014.

Cf. A081671, A098331.

T[b_,c_,0]=1;T[b_,c_,1]=b;
T[b_,c_,n_]:=T[b,c,n]=(b(2n-1)T[b,c,n-1]-(b^2-4c)(n-1)T[b,c,n-2])/n;
a[n_]:=a[n]=Sum[(40k+13)(-1)^k*50^(n-1-k)*T[4,1,k]*T[1,-1,k]^2,{k,0,n-1}]/n;
Table[a[n],{n,1,20}]

A385728 Expansion of 1/((1-2x) (1-6*x))^(3/2).

Original entry on oeis.org

1, 12, 102, 760, 5310, 35784, 235788, 1530288, 9824310, 62557000, 395797908, 2491381776, 15616141996, 97537784400, 607391245080, 3772617319008, 23379854507046, 144605546475336, 892834113930180, 5504041611527760, 33883431379007364, 208327771987901808

Offset: 0

Seiichi Manyama, Aug 19 2025

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A385563, A385813.

Cf. A005572, A081671, A331515.

R := PowerSeriesRing(Rationals(), 34); f := 1 / ((1 - 2*x) * (1 - 6*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 22 2025

Module[{a, n}, RecurrenceTable[{a[n] == ((8*n+4)*a[n-1] - 12*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 12}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)
CoefficientList[Series[ 1/((1-2*x)*(1-6*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 22 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(3/2))

n*a(n) = (8*n+4)*a(n-1) - 12*(n+1)*a(n-2) for n > 1.
a(n) = (1/2)^n * Sum_{k=0..n} 3^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A005572(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} 2^k * (-3/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(n) * 2^(n - 1/2) * 3^(n + 3/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 21 2025

A307695 Expansion of 1/(sqrt(1-4x)sqrt(1-16*x)).

Original entry on oeis.org

1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200

Offset: 0

Seiichi Manyama, Apr 22 2019

nonn

Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..831

Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).

a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)

N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))

{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}

{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}

a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 30 2019

A117852 Mirror image of A098473 formatted as a triangular array.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 20, 18, 6, 1, 70, 80, 36, 8, 1, 252, 350, 200, 60, 10, 1, 924, 1512, 1050, 400, 90, 12, 1, 3432, 6468, 5292, 2450, 700, 126, 14, 1, 12870, 27456, 25872, 14112, 4900, 1120, 168, 16, 1, 48620, 115830, 123552, 77616, 31752, 8820, 1680, 216, 18, 1

Offset: 0

Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006

			Triangle begins:
    1;
    2,   1;
    6,   4,   1;
   20,  18,   6,   1;
   70,  80,  36,   8,   1;
  252, 350, 200,  60,  10,   1;
  ...

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A098473.

c:=n->binomial(2*n, n): T:=proc(n, k) if k<=n then binomial(n, k)*c(n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; #

Table[ Binomial[n, k]*Binomial[2*n - 2*k, n - k], {n,0,10}, {k,0,n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007
T(n,k) = binomial(n,k)*A000984(n-k). - Philippe Deléham, Dec 12 2009
O.g.f.:  1/sqrt( (1 - x*t)*(1 - (x + 4)*t) ) = 1 + (2 + x)*t + (6 + 4*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 12 2007

A302181 Number of 3D walks of type abb.

Original entry on oeis.org

1, 5, 62, 1065, 21714, 492366, 12004740, 308559537, 8255788970, 227976044010, 6457854821340, 186814834574550, 5500292590186380, 164387681345290500, 4976887208815547640, 152378485941172462785, 4711642301137121933850, 146964278352052950118770, 4619875954522866283392300

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A005558, A126869.

C := n-> binomial(2*n, n)/(n+1): # Catalan numbers
A302181 := n-> add(binomial(2*n, k)*C(iquo(k+1, 2))*C(iquo(k, 2))*(2*iquo(k, 2)+1)*add((-1)^(k+j)*binomial(2*n-k, iquo(j,2)), j=0..2*n-k), k=0..2*n): seq(A302181(n), n = 0 .. 18); # Mélika Tebni, Nov 06 2024

a(n) = Sum_{k=0..2*n} binomial(2*n, k) * A005558(k) * A126869(2*n-k). - Mélika Tebni, Nov 06 2024

a(8)-a(18) from Nachum Dershowitz, Aug 03 2020

A154627 Expansion of (1-5x)/(1-8x+4x^2).

Original entry on oeis.org

1, 3, 20, 148, 1104, 8240, 61504, 459072, 3426560, 25576192, 190903296, 1424921600, 10635759616, 79386390528, 592548085760, 4422839123968, 33012520648704, 246408808693760, 1839220386955264, 13728127860867072

Offset: 0

Paul Barry, Jan 13 2009

Hankel transform of 1,1,4,18,88,.... (see A081671).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-4).

A[0]:= 1: A[1]:= 3:
for n from 2 to 100 do A[n]:= 8*A[n-1]-4*A[n-2] od:
seq(A[n],n=0..100); # Robert Israel, Aug 10 2014

LinearRecurrence[{8,-4},{1,3},20] (* Harvey P. Dale, Aug 10 2014 *)
CoefficientList[Series[(1 - 5 x)/(1 - 8 x + 4 x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 10 2014 *)

Vec((1-5*x)/(1-8*x+4*x^2)+O(x^50)) \\ Michel Marcus, Aug 10 2014

a(n) = 8*a(n-1) - 4*a(n-2), n > 1; a(0)=1, a(1)=3. [Philippe Deléham, Feb 02 2009]

A272867 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.

Original entry on oeis.org

1, 1, 4, 1, 1, 8, 18, 8, 1, 1, 12, 51, 88, 51, 12, 1, 1, 16, 100, 304, 454, 304, 100, 16, 1, 1, 20, 165, 720, 1770, 2424, 1770, 720, 165, 20, 1, 1, 24, 246, 1400, 4815, 10224, 13236, 10224, 4815, 1400, 246, 24, 1

Offset: 0

Peter Luschny, May 08 2016

			                                  1;
                            1,    4,  1;
                         1, 8,   18,  8, 1;
                    1, 12, 51,   88,  51, 12, 1;
              1, 16, 100, 304,  454,  304, 100, 16, 1;
        1, 20, 165, 720, 1770, 2424,  1770, 720, 165, 20, 1;
1, 24, 246, 1400, 4815, 10224, 13236, 10224, 4815, 1400, 246, 24, 1;

Indranil Ghosh, Rows 0..50, flattened

Cf. A005572, A081671.

T := (n,k) -> simplify(GegenbauerC(`if`(k

T[n_, k_]:=If[n<1, 1, If[kIndranil Ghosh, Apr 03 2017 *)

T(n,n) = A081671(n) for n>=0.

T(n+1,n+2)/(n+1) = A005572(n) for n>=0.

A293491 a(n) = n! * [x^n] exp((n+2)x)BesselI(0,2*x).

Original entry on oeis.org

1, 3, 18, 155, 1734, 23877, 390804, 7417377, 160256070, 3885021569, 104465601756, 3086353547433, 99399100528924, 3466411543407555, 130151205663179112, 5235127829223881895, 224609180728848273990, 10239557195235638377449, 494317596005491398892620, 25192788307121307053168673

Offset: 0

Ilya Gutkovskiy, Oct 10 2017

nonn

The n-th term of the n-th binomial transform of A000984.

N. J. A. Sloane, Transforms

Cf. A000984, A026375, A081671, A098409, A098410, A104454, A186925, A292627, A292632.

Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 x], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/Sqrt[(1 - n x) (1 - (n + 4) x)], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[Binomial[n, k] Binomial[2 k, k] n^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[(n + 2)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4/(2 + n)^2], {n, 0, 19}]

a(n) = [x^n] 1/sqrt((1 - n*x)*(1 - (n + 4)*x)).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*n^(n-k).
a(n) ~ exp(2) * BesselI(0,2) * n^n. - Vaclav Kotesovec, Oct 16 2017

A302180 Number of 3D walks of type aad.

Original entry on oeis.org

1, 1, 3, 7, 23, 71, 251, 883, 3305, 12505, 48895, 193755, 783355, 3205931, 13302329, 55764413, 236174933, 1008773269, 4343533967, 18834033443, 82201462251, 360883031291, 1592993944723, 7066748314147, 31493800133173, 140953938878821, 633354801073571, 2856369029213263

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.
Number of 3D walks of length n in the first octant using steps (1, 1, 0), (1, -1, 0), (1, 0, 1), (1, 0, -1) and (1, 0, 0) that start at the origin and end at (n, 0, 0). The analogous problem in 2D is given by the Motzkin numbers A001006. - Farzan Byramji, Mar 06 2021
Inverse binomial transform of A145867 (Number of 3D walks of type aae). - Mélika Tebni, Nov 05 2024

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A001006, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

M := n-> add(binomial(n, 2*k)*binomial(2*k, k)/(k+1), k = 0 .. iquo(n,2)): # Motzkin numbers
A302180 := n-> add((-1)^(n-k)*binomial(n, k)*add(binomial(k, j)*M(j)*M(k-j), j=0..k), k=0..n):  seq(A302180(n), n = 0 .. 26); # Mélika Tebni, Nov 05 2024

a(14)-a(26) from Farzan Byramji, Mar 06 2021

cp's OEIS Frontend

A246876 G.f.: 1 / AGM(1-12*x, sqrt((1-4*x)*(1-36*x))).

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A329073 a(n) = (1/n)*Sum_{k=0..n-1} (40k+13)*(-1)^k*50^(n-1-k)*T_k(4,1)*T_k(1,-1)^2, where T_k(b,c) denotes the coefficient of x^k in the expansion of (x^2+b*x+c)^k.

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A385728 Expansion of 1/((1-2*x) * (1-6*x))^(3/2).

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A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

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A117852 Mirror image of A098473 formatted as a triangular array.

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Maple

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A302181 Number of 3D walks of type abb.

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A154627 Expansion of (1-5x)/(1-8x+4x^2).

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A246876 G.f.: 1 / AGM(1-12x, sqrt((1-4x)(1-36x))).

A329073 a(n) = (1/n)Sum_{k=0..n-1} (40k+13)(-1)^k50^(n-1-k)T_k(4,1)T_k(1,-1)^2, where T_k(b,c) denotes the coefficient of x^k in the expansion of (x^2+bx+c)^k.

A385728 Expansion of 1/((1-2x) (1-6*x))^(3/2).

A307695 Expansion of 1/(sqrt(1-4x)sqrt(1-16*x)).

A293491 a(n) = n! * [x^n] exp((n+2)x)BesselI(0,2*x).