A014070 -id:A014070 - cp's OEIS frontend

A136463 Diagonal of square array A136462: a(n) = C((n+1)*2^(n-1), n) for n>=0.

1, 2, 15, 560, 91390, 61124064, 163995687856, 1756185841659392, 75079359427627897200, 12831653340946454374300160, 8777916355714456994772455584000, 24054320541767107204031746600673906688

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

a(n) is divisible by (n+1) for n>=0: a(n)/(n+1) = A136464(n).

Cf. A136462; A136464; rows: A136465, A014070, A136466, A101346; A136467.

Table[Binomial[(n+1)2^(n-1),n],{n,0,15}]  (* Harvey P. Dale, Apr 20 2011 *)

PARI
```
a(n)=binomial((n+1)*2^(n-1),n)
```

/* a(n) = Coefficient of x^n in series: */
a(n)=polcoeff(sum(i=0,n,((n+1)/2)^i*log(1+2^i*x +x*O(x^n))^i/i!),n)

a(n) = [x^n] Sum_{i>=0} ((n+1)/2)^i * log(1 + 2^i*x)^i/i!.
a(n) is found in row n, column 0, of matrix power A136467^(n+1) for n>=0.
a(n) ~ exp(n+1) * 2^(n*(n-1)) / sqrt(2*Pi*n). - Vaclav Kotesovec, Jul 02 2016

A136466 Row 2 of square array A136462: a(n) = C(3*2^(n-1), n) for n>=0.

Original entry on oeis.org

1, 3, 15, 220, 10626, 1712304, 927048304, 1708566412608, 10895665708319184, 244373929798154341120, 19561373281624772727757056, 5658395223117478029148167447552, 5975982733408602667847206514763365888

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

a(n) is found in row n, column 0, of matrix cube A136467^3 for n>=0.

Cf. A136462; other rows: A136465, A014070, A101346; A136467.

Table[Binomial[3*2^(n-1),n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)

PARI
```
a(n)=binomial(3*2^(n-1),n)
```

/* T(n,k) = Coefficient of x^k in series: */ a(n)=polcoeff(sum(i=0,n,(3/2)^i*log(1+2^i*x +x*O(x^n))^i/i!),n)

a(n) = [x^n] Sum_{i>=0} (3/2)^i * log(1 + 2^i*x)^i/i!.

a(n) ~ 3^n * 2^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

A136501 Triangle, read by rows, where T(n,k) = C(2^k,n-k) for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 1, 4, 1, 0, 0, 6, 8, 1, 0, 0, 4, 28, 16, 1, 0, 0, 1, 56, 120, 32, 1, 0, 0, 0, 70, 560, 496, 64, 1, 0, 0, 0, 56, 1820, 4960, 2016, 128, 1, 0, 0, 0, 28, 4368, 35960, 41664, 8128, 256, 1, 0, 0, 0, 8, 8008, 201376, 635376, 341376, 32640, 512, 1, 0, 0, 0, 1, 11440, 906192, 7624512, 10668000, 2763520, 130816, 1024, 1

Offset: 0

Paul D. Hanna, Jan 01 2008

			Triangle begins:
  1;
  1, 1;
  0, 2, 1;
  0, 1, 4,  1;
  0, 0, 6,  8,     1;
  0, 0, 4, 28,    16,      1;
  0, 0, 1, 56,   120,     32,       1;
  0, 0, 0, 70,   560,    496,      64,        1;
  0, 0, 0, 56,  1820,   4960,    2016,      128,       1;
  0, 0, 0, 28,  4368,  35960,   41664,     8128,     256,      1;
  0, 0, 0,  8,  8008, 201376,  635376,   341376,   32640,    512,    1;
  0, 0, 0,  1, 11440, 906192, 7624512, 10668000, 2763520, 130816, 1024, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A014070 (central terms), A121688 (row sums), A136502 (matrix inverse).

[Binomial(2^k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 15 2021

Table[Binomial[2^k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 15 2021 *)

PARI
```
T(n,k)=binomial(2^k,n-k)
```

flatten([[binomial(2^k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 15 2021

A136635 Triangle, read by rows, where T(n,k) = C(n,k) * C(2^k*3^(n-k), n) for n>=k>=0.

Original entry on oeis.org

1, 3, 2, 36, 30, 6, 2925, 2448, 660, 56, 1663740, 1265004, 353430, 42504, 1820, 6774333588, 4368213360, 1114691760, 139915440, 8561520, 201376, 204208594169580, 106458751541142, 23004238451040, 2630276490960

Offset: 0

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

Main diagonal is A014070(n) = C(2^n,n).
Column 0 is A136393(n) = C(3^n,n).
Row sums form A136637.
Antidiagonal sums form A136638.

			Triangle begins:
1;
3, 2;
36, 30, 6;
2925, 2448, 660, 56;
1663740, 1265004, 353430, 42504, 1820;
6774333588, 4368213360, 1114691760, 139915440, 8561520, 201376;
204208594169580, 106458751541142, 23004238451040, 2630276490960, 167150463480, 5562289824, 74974368; ...

Cf. A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136637 (row sums), A136638 (antidiagonal sums).

Flatten[Table[Binomial[n,k]Binomial[2^k 3^(n-k),n],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Dec 13 2012 *)

{T(n,k)=binomial(n,k)*binomial(2^k*3^(n-k),n)}

/* Using g.f.: */ {T(n,k)=polcoeff(polcoeff(sum(i=0,n,log(1+3^i*x+2^i*x*y)^i/i!),n,x),k,y)}

G.f.: A(x,y) = Sum_{n>=0} log(1 + 3^n*x + 2^n*x*y)^n / n!.

A136637 a(n) = Sum_{k=0..n} C(n, k) * C(2^k*3^(n-k), n).

Original entry on oeis.org

1, 5, 72, 6089, 3326498, 12405917044, 336474648380394, 69883583587428350874, 115099747754889610404191160, 1536533057081060754026861201898620, 168527150638482484315370462123098294514192

Offset: 0

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

nonn

Equals row sums of triangle A136635.

			More generally,
if Sum_{n>=0} log(1 + (p^n + r*q^n)*x )^n / n! = Sum_{n>=0} b(n)*x^n,
then b(n) = Sum_{k=0..n} C(n,k)*r^(n-k) * C(p^k*q^(n-k), n)
(a result due to _Vladeta Jovovic_, Jan 13 2008).

Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136638 (antidiagonal sums).

Table[Sum[Binomial[n,k]*Binomial[2^k*3^(n-k),n], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

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

/* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+(2^i+3^i)*x)^i/i!),n,x)}

G.f.: A(x) = Sum_{n>=0} log(1 + (2^n + 3^n)*x )^n / n!.

a(n) ~ 3^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A136638 a(n) = Sum_{k=0..[n/2]} C(n-k, k) * C(3^(n-2k)2^k, n-k).

Original entry on oeis.org

1, 3, 38, 2955, 1666194, 6775599252, 204212962736426, 47025953519744215608, 84798028785462127288681736, 1219731316443261012339196962784452, 141916030637329352970764084182705691263552

Offset: 0

Vladeta Jovovic and Paul D. Hanna, Jan 15 2008

nonn

Equals antidiagonal sums of triangle A136635.

			More generally, if Sum_{n>=0} log(1 + b*p^n*x + d*q^n*x^2)^n/n! = Sum_{n>=0} a(n)*x^n then a(n) = Sum_{k=0..[n/2]} C(n-k,k)*b^(n-2k)*d^k*C(p^(n-2k)*q^k,n-k).

Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136637 (row sums).

Table[Sum[Binomial[n-k,k]*Binomial[2^k*3^(n-2*k),n-k], {k, 0, Floor[n/2]}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

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

/* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+3^i*x+2^i*x^2)^i/i!),n,x)}

G.f.: A(x) = Sum_{n>=0} log(1 + 3^n*x + 2^n*x^2)^n / n!.

a(n) ~ 3^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A166997 G.f.: C(x)^2 - S(x)^2 where C(x) = Sum_{n>=0} log(1 - 2^(2n)x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.

Original entry on oeis.org

1, 0, 12, 56, 5404, 171664, 193729840, 36639136064, 919064160383600, 937227332865348224, 699214061851483321467008, 3577364560049979516493456896, 93123865010226899737836259608990464

Offset: 0

Paul D. Hanna, Nov 22 2009

nonn

			G.f: 1 + 12*x^2 + 56*x^3 + 5404*x^4 + 171664*x^5 + 193729840*x^6 +...
which equals C(x)^2 - S(x)^2 where
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...

Cf. A166995, A166996, A166998, A060690, A014070.

{a(n)=polcoeff(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!)^2-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!)^2,n)}

G.f.: [C(x)+S(x)]*[C(x)-S(x)] where C(x) + S(x) = g.f. of A060690 and C(-x) - S(-x) = g.f. of A014070.

Self-convolution of A166998.

A060430 Residue C(2^n,n) mod C(2^n,2).

Original entry on oeis.org

0, 0, 0, 20, 0, 1344, 0, 8160, 0, 349184, 0, 1397760, 0, 0, 306774016, 268431360, 0, 7635468288, 0, 494779760640, 942438088704, 0, 0, 105553109975040, 0, 0, 0, 6004799480791040, 0, 192153583922184192, 0, 576460752169205760, 0, 49191317527028826112, 0

Offset: 1

Labos Elemer, Apr 11 2001

Chai Wah Wu, Table of n, a(n) for n = 1..1664 (terms 1..200 from Harry J. Smith)

Cf. A014070, A006516.

for n from 1 to 60 do printf(`%d,`,binomial(2^n, n) mod binomial(2^n,2)) od:

Table[Mod[Binomial[2^n,n],Binomial[2^n,2]],{n,40}] (* Harvey P. Dale, Feb 11 2015 *)

from math import comb
def A060430(n): return comb(m:=1<Chai Wah Wu, Aug 01 2025

a(n) = A014070(n) mod A006516(n).

More terms from James Sellers, Apr 12 2001

Offset corrected by Harry J. Smith, Jul 05 2009

A060497 a(n) = binomial(2^n, prime(n)).

Original entry on oeis.org

1, 4, 56, 11440, 129024480, 13136858812224, 614965786737727286400, 23665616042362569982121184000, 4818341997044953599090397918009939136000, 150781781287041846344613980638793530715139659227762775040

Offset: 1

Labos Elemer, Apr 11 2001

			n = 4, a(4) = binomial(16,7) = binomial(2^4, p(4)) = 11440.

Harry J. Smith, Table of n, a(n) for n = 1..31

Cf. A014070, A006516.

a(n) = { binomial(2^n, prime(n)) } \\ Harry J. Smith, Jul 06 2009

A187083 a(n) = binomial(n^n, n).

Original entry on oeis.org

1, 6, 2925, 174792640, 2475588476563125, 14320984850603177651837856, 50975600425441237253196072020826978589, 155681826868802708662507744652859497547627180714885120, 541851389452483826218851027234763464912884507272826833630475746754951097

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..27
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.

Cf. A014062, A014070, A086687.

[Binomial(n^n, n): n in [1..15]]; // Vincenzo Librandi, Apr 22 2011

Mathematica
```
Table[Binomial[n^n,n],{n,12}]
```

a(n) ~ n^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

cp's OEIS Frontend

A136463 Diagonal of square array A136462: a(n) = C((n+1)*2^(n-1), n) for n>=0.

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A136466 Row 2 of square array A136462: a(n) = C(3*2^(n-1), n) for n>=0.

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A136501 Triangle, read by rows, where T(n,k) = C(2^k,n-k) for n>=k>=0.

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A136635 Triangle, read by rows, where T(n,k) = C(n,k) * C(2^k*3^(n-k), n) for n>=k>=0.

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A136637 a(n) = Sum_{k=0..n} C(n, k) * C(2^k*3^(n-k), n).

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A136638 a(n) = Sum_{k=0..[n/2]} C(n-k, k) * C(3^(n-2*k)*2^k, n-k).

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A166997 G.f.: C(x)^2 - S(x)^2 where C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.

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A136638 a(n) = Sum_{k=0..[n/2]} C(n-k, k) * C(3^(n-2k)2^k, n-k).

A166997 G.f.: C(x)^2 - S(x)^2 where C(x) = Sum_{n>=0} log(1 - 2^(2n)x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.