A014070 -id:A014070 - cp's OEIS frontend

A132684 a(n) = binomial(2^n + n + 1, n).

1, 4, 21, 220, 5985, 501942, 143218999, 145944307080, 542150225230185, 7398714129087308170, 372134605932348010322571, 69146263065062394421802892300, 47589861944854471977019273909187085

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

			From _Paul D. Hanna_, Feb 25 2009: (Start)
G.f.: A(x) = 1 + 4*x + 21*x^2 + 220*x^3 + 5985*x^4 + 501942*x^5 +...
A(x) = 1/(1-x)^2 - log(1-2x)/(1-2x)^2 + log(1-4x)^2/((1-4x)^2*2!) - log(1-8x)^3/((1-8x)^2*3!) +- ... (End)

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), this sequence (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A136555.
Cf. A066384. - Paul D. Hanna, Feb 25 2009

[Binomial(2^n +n+1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

A132684:= n-> binomial(2^n +n+1, n); seq(A132684(n), n=0..20); # G. C. Greubel, Mar 14 2021

Table[Binomial[2^n+n+1,n],{n,0,20}] (* Harvey P. Dale, Nov 10 2011 *)

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

{a(n)=polcoeff(sum(m=0,n,(-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))^2*m!)),n)} \\ Paul D. Hanna, Feb 25 2009

[binomial(2^n +n+1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

a(n) = [x^n] 1/(1-x)^(2^n + 2).
G.f.: Sum_{n>=0} (-log(1 - 2^n*x))^n / ((1 - 2^n*x)^2*n!). - Paul D. Hanna, Feb 25 2009
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A132685 a(n) = binomial(2^n + 2*n, n).

Original entry on oeis.org

1, 4, 28, 364, 10626, 850668, 218618940, 198773423848, 669741609663270, 8493008777332033900, 405943250253048290447028, 72938914603968404495709630360, 49143490709866058459392200362497820

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), this sequence (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n+2*n,n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

A132695:= n-> binomial(2^n +2*n,n); seq(A132685(n), n=0..20); # G. C. Greubel, Mar 14 2021

Table[Binomial[2^n+2n,n],{n,0,20}] (* Harvey P. Dale, Jun 01 2016 *)

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

[binomial(2^n+2*n,n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

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

A132686 a(n) = binomial(2^n + 2*n + 1, n).

Original entry on oeis.org

1, 5, 36, 455, 12650, 962598, 237093780, 209004408899, 689960224294614, 8639439963148103450, 409865407260324119340236, 73328394245057556170201283726, 49287010273876375495535472789937580

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), this sequence (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n +2*n +1, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021

Table[Binomial[2^n +2*n +1, n], {n,0,20}] (* G. C. Greubel, Mar 13 2021 *)

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

[binomial(2^n +2*n +1, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

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

A132687 a(n) = binomial(2^n + 3*n - 1, n).

Original entry on oeis.org

1, 4, 36, 560, 17550, 1370754, 324540216, 267212177232, 822871715492970, 9728874233306696390, 442491588454024774291770, 76919746769405407508866898400, 50743487119356450255156023756871000

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), this sequence (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n +3*n -1, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021

Table[Binomial[2^n+3n-1,n],{n,0,20}] (* Harvey P. Dale, Sep 07 2017 *)

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

[binomial(2^n +3*n -1, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

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

A132688 a(n) = binomial(2^n + 3*n, n).

Original entry on oeis.org

1, 5, 45, 680, 20475, 1533939, 350161812, 280384608504, 847073824772175, 9894081531608130857, 446730013630787463700695, 77328499046923986969058944720, 50891283683781760304442885961988100

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), this sequence (3,0), A132689 (3,1).

Cf. A136555.

[Binomial(2^n +3*n, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021

Table[Binomial[2^n+3n,n],{n,0,20}] (* Harvey P. Dale, Oct 30 2018 *)

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

[binomial(2^n +3*n, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021

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

A132689 a(n) = binomial(2^n + 3*n + 1, n).

Original entry on oeis.org

1, 6, 55, 816, 23751, 1712304, 377447148, 294109729200, 871896500955975, 10061777828754031380, 451004941990890693018405, 77739225019650285306412710240, 51039474754930845750609669420261300

Offset: 0

Paul D. Hanna, Aug 26 2007

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), this sequence (3,1).

Cf. A136555.

[Binomial(2^n+3*n+1, n) : n in [0..15]]; // Wesley Ivan Hurt, Nov 20 2014

A132689:=n->binomial(2^n+3*n+1, n): seq(A132689(n), n=0..15); # Wesley Ivan Hurt, Nov 20 2014

Table[Binomial[2^n +3n +1, n], {n, 0, 15}] (* Wesley Ivan Hurt, Nov 20 2014 *)

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

[binomial(2^n +3*n+1, n) for n in (0..15)] # G. C. Greubel, Feb 15 2021

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

A136393 a(n) = C(3^n,n).

Original entry on oeis.org

1, 3, 36, 2925, 1663740, 6774333588, 204208594169580, 47025847059877940202, 84798009611754271531960140, 1219731290030242386267605060168700, 141916030352038369973126553950792759280336

Offset: 0

Paul D. Hanna, Dec 28 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..45

Cf. A014070 (C(2^n, n)).

[Binomial(3^n,n): n in [0..25]]; // Vincenzo Librandi, Sep 13 2016

Table[Binomial[3^n,n], {n,0,10}] (* Vaclav Kotesovec, Jul 02 2016 *)

PARI
```
a(n)=binomial(3^n,n)
```

/* G.f. A(x) as Sum of Series: */
a(n)=polcoeff(sum(k=0,n,log(1+3^k*x +x*O(x^n))^k/k!),n)

{a(n) = (1/n!) * sum(k=0, n, stirling(n, k, 1) * 3^(n*k) )}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 05 2023

G.f.: A(x) = Sum_{n>=0} log(1 + 3^n*x)^n / n!.
a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n, k) * 3^(n*k). - Paul D. Hanna, Feb 05 2023
a(n) ~ 3^(n^2)/n!. - Vaclav Kotesovec, Jul 02 2016

A189981 E.g.f. satisfies: A(x) = Sum_{n>=0} log(1 + x*A(x)^n)^n/n!.

Original entry on oeis.org

1, 1, 2, 12, 120, 1600, 28500, 621138, 16017792, 480474720, 16390969920, 626786792280, 26584872779520, 1238524175509608, 62873918454756864, 3455537675553482400, 204449393824639488000, 12958008875933613962880

Offset: 0

Paul D. Hanna, May 03 2011

sign

The definition of the e.g.f. A(x) is an application of the identity:
* Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} binomial(q^n, n)*x^n at q = A(x).
Consider the function G(x) such that G(x) = 1 + x*G(x)^p, then
* G(x) = Sum_{n>=0} log(1 + x*G(x)^p)^n/n! (trivially), and
* G(x) = Sum_{n>=0} binomial(p*n+1,n)*x^n/(p*n+1) for fixed p;
does an analogous expression exist for the e.g.f. of this sequence?
Note that terms a(70)-a(83) are negative. - Vaclav Kotesovec, Jul 13 2014

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1600*x^5/5! +...
where A(x) satisfies:
A(x) = 1 + log(1 + x*A(x)) + log(1 + x*A(x)^2)^2/2! + log(1 + x*A(x)^3)^3/3! +...
The e.g.f. also satisfies:
A(x) = 1 + A(x)*x + A(x)^2*(A(x)^2-1)*x^2/2! + A(x)^3*(A(x)^3-1)*(A(x)^3-2)*x^3/3! + A(x)^4*(A(x)^4-1)*(A(x)^4-2)*(A(x)^4-3)*x^4/4! +...+ binomial(A(x)^n, n)*x^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..106

Cf. A014070, A221101, A224797.

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,log(1+x*(A+x*O(x^n))^m)^m/m!));n!*polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,binomial((A+x*O(x^n))^m,m)*x^m));n!*polcoeff(A,n)}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,sum(k=0,m,Stirling1(m,k)*(A+x*O(x^n))^(m*k))*x^m/m!));n!*polcoeff(A,n)}

E.g.f. also satisfies:
(1) A(x) = Sum_{n>=0} binomial(A(x)^n, n) * x^n.
(2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * A(x)^(n*k)/n!.

A136462 Square table, read by antidiagonals, where T(n,k) = C((n+1)*2^(k-1), k) for n>=0, k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 4, 1, 4, 15, 56, 70, 1, 5, 28, 220, 1820, 4368, 1, 6, 45, 560, 10626, 201376, 906192, 1, 7, 66, 1140, 35960, 1712304, 74974368, 621216192, 1, 8, 91, 2024, 91390, 7624512, 927048304, 94525795200, 1429702652400, 1, 9, 120, 3276, 194580, 24040016, 5423611200, 1708566412608, 409663695276000, 11288510714272000, 1, 10, 153, 4960, 367290, 61124064, 21193254160, 13161885792000, 10895665708319184, 6208116950265950720, 312268282598377321216

Offset: 0

Paul D. Hanna, Dec 31 2007

Row n equals column 0 of matrix product A136467^(n+1) for n>=0.

			1,1,1,4,70,4368,906192,621216192,1429702652400,11288510714272000,...;
1,2,6,56,1820,201376,74974368,94525795200,409663695276000,...;
1,3,15,220,10626,1712304,927048304,1708566412608,...;
1,4,28,560,35960,7624512,5423611200,13161885792000,...;
1,5,45,1140,91390,24040016,21193254160,63815149590720,...;
1,6,66,2024,194580,61124064,64300886496,231207760388736,...;
1,7,91,3276,367290,134153712,163995687856,685581099291712,...;
1,8,120,4960,635376,264566400,368532802176,1756185841659392,...; ...
Triangle A136467 begins:
1;
1,1;
1,4,1;
4,32,16,1;
70,848,576,64,1;
4368,75648,62208,9216,256,1;
906192,22313216,21169152,3792896,143360,1024,1;
621216192,21827627008,23212261376,4793434112,223215616,2228224,4096,1;
such that row n of A136462 equals column 0 of A136467^(n+1).

Paul D. Hanna, Table of n, a(n) for n = 0..495, for rows 0..30 of flattened table.

Cf. rows: A136465, A014070, A136466, A101346; A136463 (diagonal); A136467.

{T(n,k)=binomial((n+1)*2^(k-1),k)}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

/* T(n,k) = Coefficient of x^k in series: */
{T(n,k)=polcoeff(sum(i=0,k,((n+1)/2)^i*log(1+2^i*x +x*O(x^k))^i/i!),k)}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

O.g.f. of row n: Sum_{k>=0} ((n+1)/2)^k * log(1 + 2^k*x)^k / k! = Sum_{k>=0} C((n+1)*2^(k-1), k) * x^k for n>=0.

More terms and b-file added by Paul D. Hanna, Jul 02 2016

A136465 Row 0 of square array A136462: a(n) = C(2^(n-1), n) for n>=0.

Original entry on oeis.org

1, 1, 1, 4, 70, 4368, 906192, 621216192, 1429702652400, 11288510714272000, 312268282598377321216, 30813235422145714150738944, 11005261717918037175659349191168, 14391972654784168932973746746691440640, 69538271351155829150354851003285125277716480, 1250303357941919088313448625534941836891635347865600

Offset: 0

Paul D. Hanna, Dec 31 2007

a(n) is found in row n, column 0, of triangle A136467 for n>=0.
For n > 0, number of increasing integer sequences 1 <= a_1 < ... < a_n <= 2^(n-1). - Charles R Greathouse IV, Aug 08 2010
The (n-1)-dimensional hypercube has 2^(n-1) corners.  There are binomial(2^(n-1),n) ways of selecting a set of n corners. So a(n) is the number of simplices (hyper-tetrahedra) with vertices defined by a corner subset of a (n-1)-dimensional hypercube. (This count includes degenerate polytopes with zero volume.) - R. J. Mathar, Jan 16 2016

			From _Paul D. Hanna_, Sep 26 2010: (Start)
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 70*x^4 + 4368*x^5 +...
A(x) = 1 + log(1+2*x)/2 + log(1 + 2^2*x)^2/(2!*2^2) + log(1 + 2^3*x)^3/(3!*2^3) + log(1 + 2^4*x)^4/(4!*2^4) +... (End)

J. Brandts, S. Dijkhuis, V. de Haan, M. Krizek, There are only two nonobtuse binary triangulations of the unit n-cube, Comp. Geom. 46 (2013) 286-297, Table 1.

Cf. A136462; other rows: A014070, A136466, A101346; A136467.

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

{a(n)=binomial(2^(n-1),n)}
for(n=0,20,print1(a(n),", "))

/* a(n) = Coefficient of x^k in series: */
{a(n)=polcoeff(sum(i=0,n,(1/2)^i*log(1+2^i*x +x*O(x^n))^i/i!),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=polcoeff(sum(m=0,n,log(1+2^m*x+x*O(x^n))^m/(m!*2^m)),n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Sep 26 2010

a(n) = [x^n] Sum_{i>=0} (1/2)^i * log(1 + 2^i*x)^i/i!.
O.g.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n!*2^n). - Paul D. Hanna, Sep 26 2010
a(n) ~ 2^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

cp's OEIS Frontend

A132684 a(n) = binomial(2^n + n + 1, n).

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A132685 a(n) = binomial(2^n + 2*n, n).

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A132686 a(n) = binomial(2^n + 2*n + 1, n).

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A132687 a(n) = binomial(2^n + 3*n - 1, n).

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A132688 a(n) = binomial(2^n + 3*n, n).

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Mathematica

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A132689 a(n) = binomial(2^n + 3*n + 1, n).

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A136393 a(n) = C(3^n,n).

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