A060690 -id:A060690 - cp's OEIS frontend

A140051 L.g.f.: A(x) = log(G(x)) where G(x) = g.f. of A060690(n) = C(2^n+n-1,n).

2, 16, 308, 14488, 1843232, 714580528, 917085102992, 4076698622618144, 64300718807613519968, 3649606003781552269341376, 752497581806524062754828125952, 567745591696108934746387351412913664

Offset: 1

Paul D. Hanna, May 02 2008

nonn

			A(x) = 2*x + 16*x^2/2 + 308*x^3/3 + 14488*x^4/4 + 1843232*x^5/5 +...
A(x) = log(G(x)) where G(x) = g.f. of A060690:
G(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 +... + C(2^n+n-1,n)*x^n +...

Cf. A140050, A060690.

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

L.g.f.: A(x) = log[ Sum_{n>=0} (-log(1 - 2^n*x))^n/n! ].

A014070 a(n) = binomial(2^n, n).

Original entry on oeis.org

1, 2, 6, 56, 1820, 201376, 74974368, 94525795200, 409663695276000, 6208116950265950720, 334265867498622145619456, 64832175068736596027448301568, 45811862025512780638750907861652480, 119028707533461499951701664512286557511680

Offset: 0

N. J. A. Sloane

a(n) is the number of n X n (0,1) matrices with distinct rows modulo rows permutations. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), this sequence (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), A132689 (3,1).

Cf. A088229, A136393, A136555.

[Binomial(2^n, n): n in [0..25]]; // Vincenzo Librandi, Sep 13 2016

A014070:= n-> binomial(2^n,n); seq(A014070(n), n=0..20); # G. C. Greubel, Mar 14 2021

Table[Binomial[2^n,n],{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)

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

/* G.f. A(x) as Sum of Series: */
a(n)=polcoeff(sum(k=0,n,log(1+2^k*x +x*O(x^n))^k/k!),n) \\ Paul D. Hanna, Dec 28 2007

{a(n) = (1/n!) * sum(k=0,n, stirling(n, k, 1) * 2^(n*k) )}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Feb 05 2023

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

G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n / n!. - Paul D. Hanna, Dec 28 2007
a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n, k) * 2^(n*k). - Paul D. Hanna, Feb 05 2023
From Vaclav Kotesovec, Jul 02 2016: (Start)
a(n) ~ 2^(n^2) / n!.
a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).
(End)

A136556 a(n) = binomial(2^n - 1, n).

Original entry on oeis.org

1, 1, 3, 35, 1365, 169911, 67945521, 89356415775, 396861704798625, 6098989894499557055, 331001552386330913728641, 64483955378425999076128999167, 45677647585984911164223317311276545, 118839819203635450208125966070067352769535, 1144686912178270649701033287538093722740144666625

Offset: 0

Paul D. Hanna, Jan 07 2008; Paul Hanna and Vladeta Jovovic, Jan 15 2008

nonn

Number of n x n binary matrices without zero rows and with distinct rows up to permutation of rows, cf. A014070.
Row 0 of square array A136555.
From Gus Wiseman, Dec 19 2023: (Start)
Also the number of n-element sets of nonempty subsets of {1..n}, or set-systems with n vertices and n edges (not necessarily covering). The covering case is A054780. For example, the a(3) = 35 set-systems are:
  {1}{2}{3}  {1}{2}{12}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{13}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{2}{23}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{12}  {1}{12}{23}  {2}{12}{123}
             {1}{3}{13}  {1}{13}{23}  {2}{13}{123}
             {1}{3}{23}  {2}{3}{123}  {2}{23}{123}
             {2}{3}{12}  {2}{12}{13}  {3}{12}{123}
             {2}{3}{13}  {2}{12}{23}  {3}{13}{123}
             {2}{3}{23}  {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
Of these, only {{1},{2},{1,2}}, {{1},{3},{1,3}}, and {{2},{3},{2,3}} do not cover the vertex set.
(End)

			G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 1365*x^4 + 169911*x^5 +...
A(x) = 1/(1+x) + log(1+2*x)/(1+2*x) + log(1+4*x)^2/(2!*(1+4*x)) + log(1+8*x)^3/(3!*(1+8*x)) + log(1+16*x)^4/(4!*(1+16*x)) + log(1+32*x)^5/(5!*(1+32*x)) +...

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

Sequences of the form binomial(2^n +p*n +q, n): this sequence (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), A132689 (3,1).
Cf. A066384, A136555, A136557.
The covering case A054780 has binomial transform A367916, ranks A367917.
Connected graphs of this type are A057500, unlabeled A001429.
Graphs of this type are A116508, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A058891 counts set-systems, connected A323818, without singletons A016031.
Cf. A055154, A059201, A133686, A367862, A367902, A367903.

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

A136556:= n-> binomial(2^n-1,n); seq(A136556(n), n=0..20); # G. C. Greubel, Mar 14 2021

f[n_] := Binomial[2^n - 1, n]; Array[f, 12] (* Robert G. Wilson v *)
Table[Length[Subsets[Rest[Subsets[Range[n]]],{n}]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)

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

/* As coefficient of x^n in the g.f.: */
{a(n) = polcoeff( sum(i=0,n, 1/(1 + 2^i*x +x*O(x^n)) * log(1 + 2^i*x +x*O(x^n))^i/i!), n)}
for(n=0, 20, print1(a(n), ", "))

from math import comb
def A136556(n): return comb((1<Chai Wah Wu, Jan 02 2024

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2^n,k).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k) * (2^n-1)^k.
G.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n! * (1 + 2^n*x)).
a(n) ~ 2^(n^2)/n!. - Vaclav Kotesovec, Jul 02 2016

Edited by N. J. A. Sloane, Jan 26 2008

A016121 Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

Original entry on oeis.org

1, 2, 5, 17, 86, 698, 9551, 226592, 9471845, 705154187, 94285792211, 22807963405043, 10047909839840456, 8110620438438750647, 12062839548612627177590, 33226539134943667506533207, 170288915434579567358828997806, 1630770670148598007261992936663653

Offset: 0

Jeffrey Shallit

nonn

Number of n X n binary symmetric matrices with rows, considered as binary numbers, in nondecreasing order. - R. H. Hardin, May 30 2008

Also, number of (n+1) X (n+1) binary symmetric matrices with zero main diagonal and rows, considered as binary numbers, in nondecreasing order. - Max Alekseyev, Feb 06 2022

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A008934, A060690, A089006, A351157, A351158, A351287, A351288.

Row sums of triangle A097712.

T[n_, k_] := T[n, k] = If[n < 0 || k > n, 0, If[n == k, 1, If[k == 0, 1, T[n - 1, k] + Sum[T[n - 1, j] T[j, k - 1], {j, 0, n - 1}]]]];
a[n_] := Sum[T[n, k], {k, 0, n}];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 02 2019 *)

@CachedFunction
def T(n, k): # T = A097712
    if k<0 or k>n: return 0
    elif k==0 or k==n: return 1
    else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
def A016121(n): return sum(T(n,k) for k in range(n+1))
[A016121(n) for n in range(31)] # G. C. Greubel, Feb 21 2024

a(n) = Sum_{k=0..n} A097712(n, k). - Paul D. Hanna, Aug 24 2004

Equals the binomial transform of A008934 (number of tournament sequences): a(n) = Sum_{k=0..n} C(n, k)*A008934(k). - Paul D. Hanna, Sep 18 2005

A136555 Square array, read by antidiagonals, where T(n,k) = binomial(2^k + n-1, k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 35, 1, 4, 10, 56, 1365, 1, 5, 15, 84, 1820, 169911, 1, 6, 21, 120, 2380, 201376, 67945521, 1, 7, 28, 165, 3060, 237336, 74974368, 89356415775, 1, 8, 36, 220, 3876, 278256, 82598880, 94525795200, 396861704798625, 1, 9, 45, 286, 4845, 324632, 90858768, 99949406400, 409663695276000, 6098989894499557055

Offset: 0

Paul D. Hanna, Jan 07 2008

Let vector R_{n} equal row n of this array; then R_{n+1} = P * R_{n} for n>=0, where triangle P = A132625 such that row n+1 of P = row n of P^(2^n) with appended '1' for n>=0.

			Square array begins:
  1, 1,  3,  35, 1365, 169911,  67945521,  89356415775, ... A136556;
  1, 2,  6,  56, 1820, 201376,  74974368,  94525795200, ... A014070;
  1, 3, 10,  84, 2380, 237336,  82598880,  99949406400, ... A136505;
  1, 4, 15, 120, 3060, 278256,  90858768, 105637584000, ... A136506;
  1, 5, 21, 165, 3876, 324632,  99795696, 111600996000, ... ;
  1, 6, 28, 220, 4845, 376992, 109453344, 117850651776, ... ;
  1, 7, 36, 286, 5985, 435897, 119877472, 124397910208, ... ;
  1, 8, 45, 364, 7315, 501942, 131115985, 131254487936, ... ;
  ...
Form column vector R_{n} out of row n of this array;
then row n+1 can be generated from row n by:
R_{n+1} = P * R_{n} for n>=0,
where triangular matrix P = A132625 begins:
        1;
        1,      1;
        2,      1,     1;
       14,      4,     1,    1;
      336,     60,     8,    1,  1;
    25836,   2960,   248,   16,  1, 1;
  6251504, 454072, 24800, 1008, 32, 1, 1; ...
where row n+1 of P = row n of P^(2^n) with appended '1' for n>=0.

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Rows: A014070, A136505, A136506, A136556.
Diagonals: A060690, A132683, A132684.
Cf. A136557 (antidiagonal sums).
Cf. A132625.

[Binomial(2^k +n-k-1, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 14 2021

A136555:= (n,k) -> binomial(2^k +n-k-1, k); seq(seq(A136555(n,k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2021

Table[Binomial[2^k +n-k-1, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 14 2021 *)

PARI
```
T(n,k)=binomial(2^k+n-1,k)
```

/* Coefficient of x^k in g.f. of row n: */ T(n,k)=polcoeff(sum(i=0,k,(1+2^i*x+x*O(x^k))^(n-1)*log((1+2^i*x)+x*O(x^k))^i/i!),k)

flatten([[binomial(2^k +n-k-1, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 14 2021

G.f. for row n: Sum_{i>=0} (1 + 2^i*x)^(n-1) * log(1 + 2^i*x)^i / i!.
From G. C. Greubel, Mar 14 2021: (Start)
For the square array:
    T(n, n) = A060690(n).
  T(n+1, n) = A132683(n),   T(n+2, n) = A132684(n).
T(2*n+1, n) = A132685(n),   T(2*n, n) = A132686(n).
T(3*n+2, n) = A132689(n), T(3*n+1, n) = A132688(n), T(3*n, n) = A132687(n).
For the number triangle:
t(n, k) = T(n-k, k) = binomial(2^k + n - k -1, k).
Sum_{k=0..n} t(n,k) = Sum_{k=0..n} T(n-k, k) = A136557(n). (End)

A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

Original entry on oeis.org

1, 2, 3, 10, 5, 36, 7, 120, 45, 100, 11, 936, 13, 196, 225, 3876, 17, 3078, 19, 4200, 441, 484, 23, 62400, 325, 676, 3654, 11368, 29, 27000, 31, 376992, 1089, 1156, 1225, 443556, 37, 1444, 1521, 459200, 41, 74088, 43, 43560, 46575, 2116, 47, 11995200, 1225

Offset: 1

Paul D. Hanna, Aug 04 2009

nonn

Also the number of length n - 1 chains of divisors of n. - Gus Wiseman, May 07 2021

			Successive Dirichlet self-convolutions of the all 1's sequence begin:
(1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... (A000012)
1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)
1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)
1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)
1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)
1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)
1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)
1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)
1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)
1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)
1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)
1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)
...
where the main diagonal forms this sequence.
From _Gus Wiseman_, May 07 2021: (Start)
The a(1) = 1 through a(5) = 5 chains of divisors:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)
      (2)  (3/1)  (2/1/1)  (5/1/1/1)
           (3/3)  (2/2/1)  (5/5/1/1)
                  (2/2/2)  (5/5/5/1)
                  (4/1/1)  (5/5/5/5)
                  (4/2/1)
                  (4/2/2)
                  (4/4/1)
                  (4/4/2)
                  (4/4/4)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Enrique Pérez Herrero)

Main diagonal of A077592.
Diagonal n = k + 1 of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005 counts divisors.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts nonempty strict chains of divisors of n.
A251683/A334996 count strict nonempty length-k divisor chains from n to 1.
A337255 counts strict length-k chains of divisors starting with n.
A339564 counts factorizations with a selected factor.
A343662 counts strict length-k chains of divisors (row sums: A337256).
Cf. A002033, A007425, A008480, A018818, A062319, A066959, A186972, A327527, A337105, A337107, A343658.
Cf. A060690.

Table[Times@@(Binomial[#+n-1,n-1]&/@FactorInteger[n][[All,2]]),{n,1,50}] (* Enrique Pérez Herrero, Dec 25 2013 *)

{a(n,m=n)=if(n==1,1,if(m==1,1,sumdiv(n,d,a(d,1)*a(n/d,m-1))))}

from math import prod, comb
from sympy import factorint
def A163767(n): return prod(comb(n+e-1,e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024

a(p) = p for prime p.
a(n) = n^k when n is the product of k distinct primes (conjecture).
a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.
a(2^n) = A060690(n). - Alois P. Heinz, Jun 12 2024

A136505 a(n) = binomial(2^n + 1, n).

Original entry on oeis.org

1, 3, 10, 84, 2380, 237336, 82598880, 99949406400, 422825581068000, 6318976181520699840, 337559127276933693852160, 65182103393445184131620004864, 45946437874792132748338425828443136

Offset: 0

Paul D. Hanna, Jan 01 2008

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), this sequence (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), A132689 (3,1).

Cf. A136507, A136555.

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

A136505:= n-> binomial(2^n+1,n); seq(A136505(n), n=0..20); # G. C. Greubel, Mar 14 2021

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

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

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

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

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

A136506 a(n) = binomial(2^n + 2, n).

Original entry on oeis.org

1, 4, 15, 120, 3060, 278256, 90858768, 105637584000, 436355999662176, 6431591598617108352, 340881559632021623909760, 65533747894341651530074060800, 46081376018330435634530315478453248

Offset: 0

Paul D. Hanna, Jan 01 2008

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), this sequence (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136507, A136555.

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

A136506:= n-> binomial(2^n+2,n); seq(A136506(n), n=0..20); # G. C. Greubel, Mar 14 2021

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

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

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

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

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

A094223 Number of binary n X n matrices with all rows (columns) distinct, up to permutation of columns (rows).

Original entry on oeis.org

1, 2, 7, 68, 2251, 247016, 89254228, 108168781424, 451141297789858, 6625037125817801312, 348562672319990399962384, 66545827618461283102105245248, 46543235997095840080425299916917968, 120155975713532210671953821005746669185792, 1152009540439950050422144845158703009569109376384

Offset: 0

Goran Kilibarda and Vladeta Jovovic, May 28 2004

Andrew Howroyd, Table of n, a(n) for n = 0..50
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Main diagonal of A059584 and A059587, A060690, A088309.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

a[n_] := Sum[(-1)^(n - k)*StirlingS1[n, k]*Binomial[2^k, n], {k, 0, n}]; (* or *) a[n_] := Sum[ StirlingS1[n, k]*Binomial[2^k + n - 1, n], {k, 0, n}]; Table[ a[n], {n, 0, 12}] (* Robert G. Wilson v, May 29 2004 *)

a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k+n-1, n)); \\ Michel Marcus, Dec 17 2022

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*binomial(2^k, n).

a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k+n-1, n).

More terms from Robert G. Wilson v, May 29 2004

a(13) onwards from Andrew Howroyd, Jan 20 2024

A132683 a(n) = binomial(2^n + n, n).

Original entry on oeis.org

1, 3, 15, 165, 4845, 435897, 131115985, 138432467745, 525783425977953, 7271150092378906305, 368539102493388126164865, 68777035446753808820521420545, 47450879627176629761462147774626305

Offset: 0

Paul D. Hanna, Aug 26 2007

nonn

			From _Paul D. Hanna_, Feb 25 2009: (Start)
G.f.: A(x) = 1 + 3*x + 15*x^2 + 165*x^3 + 4845*x^4 + 435897*x^5 + ...
A(x) = 1/(1-x) - log(1-2x)/(1-2x) + log(1-4x)^2/((1-4x)*2!) - log(1-8x)^3/((1-8x)*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), this sequence (1,0), A132684 (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, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

A132683:= n-> binomial(2^n +n,n); seq(A132683(n), n=0..20); # G. C. Greubel, Mar 14 2021

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

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

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

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

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

cp's OEIS Frontend

A140051 L.g.f.: A(x) = log(G(x)) where G(x) = g.f. of A060690(n) = C(2^n+n-1,n).

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A014070 a(n) = binomial(2^n, n).

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A136556 a(n) = binomial(2^n - 1, n).

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A016121 Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

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A136555 Square array, read by antidiagonals, where T(n,k) = binomial(2^k + n-1, k).

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A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

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