A063886 -id:A063886 - cp's OEIS frontend

A303538 Expansion of ((1 + 8x)/(1 - 8x))^(1/8).

1, 2, 2, 44, 86, 1724, 4244, 80024, 223718, 4033132, 12260988, 213418728, 689489148, 11663520216, 39489621864, 652201870896, 2292944058246, 37099981422156, 134565259916012, 2138626858270408, 7964821656989332, 124595233474799752, 474734644904361112

Offset: 0

Seiichi Manyama, Apr 25 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A063886, A303537.

a[n]:=if n<2 then n+1 else (2*a[n-1]+64*(n-2)*a[n-2])/n;
makelist(a[n],n,0,1000); /* Tani Akinari, Apr 29 2018 */

N=66; x='x+O('x^N); Vec(((1+8*x)/(1-8*x))^(1/8))

a(n) ~ 2^(3*n + 1/8) / (Gamma(1/8) * n^(7/8)). - Vaclav Kotesovec, Apr 26 2018

n*a(n)-2*a(n-1)-64*(n-2)*a(n-2)=0. - Tani Akinari, Apr 29 2018

A360321 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 16, 130, 1070, 8902, 74724, 631902, 5376840, 45990070, 395106656, 3407196982, 29477061166, 255733684010, 2224098916300, 19384492018770, 169270624419390, 1480625235653670, 12970844831940000, 113785067475668550, 999400688480388570

Offset: 0

Seiichi Manyama, Feb 03 2023

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A063886, A085362, A360317, A360318, A360319, A360322.

Table[Sum[5^(n-k) Binomial[n-1,n-k]Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jun 22 2025 *)

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

my(N=30, x='x+O('x^N)); Vec(sqrt((1-5*x)/(1-9*x)))

G.f.: sqrt( (1-5*x)/(1-9*x) ).
n*a(n) = 2*(7*n-6)*a(n-1) - 45*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/5)^i * a(j) * a(i-j) = (9/5)^n.
a(n) ~ 2 * 3^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 15, 21, 27, 38, 47, 63, 79, 106, 130, 170, 209, 272, 330, 422, 512, 653, 784, 986, 1183, 1482, 1765, 2191, 2604, 3218, 3804, 4666, 5504, 6726, 7898, 9592, 11240, 13602, 15880, 19122, 22277, 26733, 31048, 37102, 43003, 51232, 59220

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(7) = 8 partitions: (7), (511), (421), (331), (322), (31111), (22111), (1111111). Missing are: (61), (52), (43), (4111), (3211), (2221), (211111).

Even- and odd-indexed terms are A006330 and A001523 respectively, which add up to A000712.

Cf. A000898, A001405, A026010, A045931, A058696, A063886, A097613, A130780, A171966, A239241, A300788, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&]],{n,0,50}]

A325699 Number of distinct even prime indices of n minus the number of distinct odd prime indices of n.

Original entry on oeis.org

0, -1, 1, -1, -1, 0, 1, -1, 1, -2, -1, 0, 1, 0, 0, -1, -1, 0, 1, -2, 2, -2, -1, 0, -1, 0, 1, 0, 1, -1, -1, -1, 0, -2, 0, 0, 1, 0, 2, -2, -1, 1, 1, -2, 0, -2, -1, 0, 1, -2, 0, 0, 1, 0, -2, 0, 2, 0, -1, -1, 1, -2, 2, -1, 0, -1, -1, -2, 0, -1, 1, 0, -1, 0, 0, 0

Offset: 1

Gus Wiseman, May 17 2019

sign

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

a(n) = A324967(n) - A324966(n).

Cf. A000712, A001221, A026010, A045931, A063886, A112798, A130780, A171966, A241638, A325698, A325700.

Table[Total[(-1)^PrimePi/@First/@If[n==1,{},FactorInteger[n]]],{n,100}]

G.f.: Sum_{k>=1} (-1)^k * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 12 2020

Additive with a(p^e) = (-1)^primepi(p). - Amiram Eldar, Jun 17 2024

A360319 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, 14, 100, 726, 5340, 39692, 297544, 2245990, 17050796, 130061412, 996078456, 7654571772, 58995989400, 455857911768, 3530234227344, 27392392806534, 212918339726028, 1657570714812020, 12922254685161112, 100867892292766612

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360318, A360321, A360322.

a(n) = sum(k=0, n, 4^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

my(N=30, x='x+O('x^N)); Vec(sqrt((1-4*x)/(1-8*x)))

G.f.: sqrt( (1-4*x)/(1-8*x) ).
n*a(n) = 2*(6*n-5)*a(n-1) - 32*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/4)^i * a(j) * a(i-j) = 2^n.
a(n) ~ 2^(3*n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 8^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

Original entry on oeis.org

1, 2, -4, 10, -30, 102, -376, 1462, -5900, 24470, -103644, 446382, -1948854, 8605290, -38362200, 172423770, -780496110, 3554991270, -16281079900, 74927379550, -346328465930, 1607078948690, -7483861047480, 34963419415650, -163825013554400, 769694347677002

Offset: 0

Seiichi Manyama, Feb 03 2023

Cf. A063886, A085362, A360317, A360318, A360319, A360321.

Cf. A007317.

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

my(N=30, x='x+O('x^N)); Vec(sqrt((1+5*x)/(1+x)))

G.f.: sqrt( (1+5*x)/(1+x) ).
n*a(n) = 2*(-3*n+4)*a(n-1) - 5*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (-1/5)^i * a(j) * a(i-j) = (1/5)^n.
a(n) = 2 * (-1)^(n+1) * A007317(n) for n > 0.
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (-1/4)^n * Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = (-1)^n * Sum_{k=0..n} binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A182027 a(n) = number of n-lettered words in the alphabet {1, 2} with as many occurrences of the substring (consecutive subword) [1, 1] as of [2, 2].

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 12, 20, 40, 70, 140, 252, 504, 924, 1848, 3432, 6864, 12870, 25740, 48620, 97240, 184756, 369512, 705432, 1410864, 2704156, 5408312, 10400600, 20801200, 40116600, 80233200, 155117520, 310235040, 601080390, 1202160780, 2333606220, 4667212440, 9075135300, 18150270600, 35345263800

Offset: 0

N. J. A. Sloane, Apr 07 2012

nonn

Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv preprint arXiv:1112.6207, 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence. [From _N. J. A. Sloane_, Apr 07 2012]

Apart from initial terms, same as A063886.

a:= proc(n) option remember; `if`(n<3, [1,2$3][n+1],
      (2*a(n-1)+4*(n-3)*a(n-2))/(n-1))
    end:
seq(a(n), n=0..39);  # Alois P. Heinz, May 11 2024

G.f.: 1 + x + x*sqrt((1+2*x)/(1-2*x))= 1 + x + x/G(0), where G(k)= 1 - 2*x/(1 + 2*x/(1 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013

E.g.f.: 1 + x - (x*BesselI(1, 2*x)*(2 + Pi*(1 + 2*x)*StruveL(0, 2*x)) - x*(1 + 2*x)*BesselI(0, 2*x)*(2 + Pi*StruveL(1, 2*x)))/2. - Stefano Spezia, May 11 2024

A305031 Expansion of ((1 + 2x)/(1 - 2x))^(3/2).

Original entry on oeis.org

1, 6, 18, 44, 102, 228, 500, 1080, 2310, 4900, 10332, 21672, 45276, 94248, 195624, 404976, 836550, 1724580, 3549260, 7293000, 14965236, 30669496, 62783448, 128388624, 262303132, 535422888, 1092063000, 2225728400, 4533175800, 9226818000, 18769219920, 38158909920

Offset: 0

Seiichi Manyama, May 24 2018

nonn

Let ((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ... then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..3000

((1 + 2*x)/(1 - 2*x))^(m/2): A063886 (m=1), this sequence (m=3), A241204 (m=4).

Cf. A304941, A305032.

[n le 2 select 6^(n-1) else 2*(3*Self(n-1) + 2*(n-3)*Self(n-2))/(n-1): n in [1..40]]; // G. C. Greubel, Jun 07 2023

CoefficientList[Series[((1+2*x)/(1-2*x))^(3/2), {x,0,40}], x] (* G. C. Greubel, Jun 07 2023 *)

N=66; x='x+O('x^N); Vec(((1+2*x)/(1-2*x))^(3/2))

@CachedFunction
def a(n): # b = A305031
    if n<2: return 6^n
    else: return 2*(3*a(n-1) + 2*(n-2)*a(n-2))//n
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

n*a(n) = 6*a(n-1) + 4*(n-2)*a(n-2) for n > 1.

a(n) ~ 2^(n + 5/2) * sqrt(n/Pi). - Vaclav Kotesovec, May 28 2018

A092266 Expansion of (1+4x)/AGM(1+4x,1-4x) where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

1, 4, 4, 16, 36, 144, 400, 1600, 4900, 19600, 63504, 254016, 853776, 3415104, 11778624, 47114496, 165636900, 662547600, 2363904400, 9455617600, 34134779536, 136539118144, 497634306624, 1990537226496, 7312459672336

Offset: 0

Michael Somos, Feb 16 2004

nonn

CoefficientList[Series[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x)), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2019 *)

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

Vec( 1/agm(1,(1-4*x)/(1+4*x)+O(x^66)) ) \\ Joerg Arndt, Aug 14 2013

G.f.: (1+4x)/AGM(1+4x, 1-4x) where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.
a(n) = A063886(n)^2.
a(2n) = A002894(n); a(2n+1) = 4*a(2n).
a(n) ~ 2^(2*n + 1) / (Pi*n). - Vaclav Kotesovec, Sep 27 2019

A304940 Expansion of ((1 + 4x)/(1 - 4x))^(1/2).

Original entry on oeis.org

1, 4, 8, 32, 96, 384, 1280, 5120, 17920, 71680, 258048, 1032192, 3784704, 15138816, 56229888, 224919552, 843448320, 3373793280, 12745441280, 50981765120, 193730707456, 774922829824, 2958796259328, 11835185037312, 45368209309696, 181472837238784

Offset: 0

Seiichi Manyama, May 22 2018

nonn

Let ((1 + k*x)/(1 - k*x))^(m/k) = a(0) + a(1)*x + a(2)*x^2 + ...

Then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

((1 + 4*x)/(1 - 4*x))^(m/4): A303537 (m=1), this sequence (m=2), A304941 (m=3), A081654 (m=4).

Cf. A063886.

N=66; x='x+O('x^N); Vec(((1+4*x)/(1-4*x))^(1/2))

n*a(n) = 4*a(n-1) + 4^2*(n-2)*a(n-2) for n > 1.

a(n) = 2^n * A063886(n).

cp's OEIS Frontend

A303538 Expansion of ((1 + 8*x)/(1 - 8*x))^(1/8).

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A360321 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.

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A325699 Number of distinct even prime indices of n minus the number of distinct odd prime indices of n.

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A360319 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

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A182027 a(n) = number of n-lettered words in the alphabet {1, 2} with as many occurrences of the substring (consecutive subword) [1, 1] as of [2, 2].

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A305031 Expansion of ((1 + 2*x)/(1 - 2*x))^(3/2).

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A092266 Expansion of (1+4x)/AGM(1+4x,1-4x) where AGM denotes the arithmetic-geometric mean.

A303538 Expansion of ((1 + 8x)/(1 - 8x))^(1/8).

A305031 Expansion of ((1 + 2x)/(1 - 2x))^(3/2).

A304940 Expansion of ((1 + 4x)/(1 - 4x))^(1/2).