A092056 -id:A092056 - cp's OEIS frontend

A060690 a(n) = binomial(2^n + n - 1, n).

1, 2, 10, 120, 3876, 376992, 119877472, 131254487936, 509850594887712, 7145544812472168960, 364974894538906616240640, 68409601066028072105113098240, 47312269462735023248040155132636160, 121317088003402776955124829814219234385920

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001

nonn

Also the number of n X n (0,1) matrices modulo rows permutation (by symmetry this is the same as the number of (0,1) matrices modulo columns permutation), i.e., the number of equivalence classes where two matrices A and B are equivalent if one of them is the result of permuting the rows of the other. The total number of (0,1) matrices is in sequence A002416.

Row sums of A220886. - Geoffrey Critzer, Nov 20 2014

Harry J. Smith, Table of n, a(n) for n = 0..59

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), this sequence (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A002416, A060336, A088309, A163767.
Cf. A136555, A220886.
Main diagonal of A092056.
Central terms of A137153.

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

with(combinat): for n from 0 to 20 do printf(`%d,`,binomial(2^n+n-1, n)) od:

Table[Binomial[2^n+n-1,n],{n,0,20}] (* Harvey P. Dale, Apr 19 2012 *)

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

{a(n)=polcoeff(sum(k=0,n,(-log(1-2^k*x +x*O(x^n)))^k/k!),n)} \\ Paul D. Hanna, Dec 29 2007

a(n) = sum(k=0, n, stirling(n,k,1)*(2^n+n-1)^k/n!); \\ Paul D. Hanna, Nov 20 2014

from math import comb
def A060690(n): return comb((1<Chai Wah Wu, Jul 05 2024

[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).
a(n) = (1/n!)*Sum_{k=0..n} ( (-1)^(n-k)*Stirling1(n, k)*2^(k*n) ). - Vladeta Jovovic, May 28 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2^n+n,k) - Vladeta Jovovic, Jan 21 2008
a(n) = Sum_{k=0..n} Stirling1(n,k)*(2^n+n-1)^k/n!. - Vladeta Jovovic, Jan 21 2008
G.f.: A(x) = Sum_{n>=0} [ -log(1 - 2^n*x)]^n / n!. More generally, Sum_{n>=0} [ -log(1 - q^n*x)]^n/n! = Sum_{n>=0} C(q^n+n-1,n)*x^n ; also Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} C(q^n,n)*x^n. - Paul D. Hanna, Dec 29 2007
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016
a(n) = A163767(2^n). - Alois P. Heinz, Jun 12 2024

More terms from James Sellers, Apr 20 2001

Edited by N. J. A. Sloane, Mar 17 2008

A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 8, 1, 1, 5, 20, 36, 16, 1, 1, 6, 35, 120, 136, 32, 1, 1, 7, 56, 330, 816, 528, 64, 1, 1, 8, 84, 792, 3876, 5984, 2080, 128, 1, 1, 9, 120, 1716, 15504, 52360, 45760, 8256, 256, 1, 1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896

Offset: 0

Paul D. Hanna, Jan 24 2008

Matrix inverse is A137156.

T(n,k) is the number of relations between a set of k distinguishable elements and a set of n-k indistinguishable elements. - Isaac R. Browne, Jun 04 2025

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   4,    1;
  1,  4,  10,    8,     1;
  1,  5,  20,   36,    16,      1;
  1,  6,  35,  120,   136,     32,      1;
  1,  7,  56,  330,   816,    528,     64,      1;
  1,  8,  84,  792,  3876,   5984,   2080,    128,     1;
  1,  9, 120, 1716, 15504,  52360,  45760,   8256,   256,   1;
  1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896, 512, 1;
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080; rows 0..45 of flattened triangle.

Cf. A137154 (row sums), A137155 (antidiagonal sums), A060690 (central terms); A137156 (matrix inverse).

Cf. A092056 (same with reflected rows).

Table[Binomial[2^k+n-k-1,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 06 2017 *)

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

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

A092055 a(n) = binomial(2 + 2^n,3).

Original entry on oeis.org

1, 4, 20, 120, 816, 5984, 45760, 357760, 2829056, 22500864, 179481600, 1433753600, 11461636096, 91659526144, 733141975040, 5864598896640, 46914643623936, 375308558925824, 3002434111406080, 24019335451770880, 192154133857304576, 1537230871833083904, 12297838178567454720

Offset: 0

Henry Bottomley, Feb 19 2004

a(n) = Sum_{i=1...(2^n)} i*(i+1)/2, this sequence is thus similar to A016131 as it is a sum of triangular numbers on the interval <1,2^n>, A016131 is a sum of triangular numbers on the interval <1,2^n - 1>. - Ctibor O. Zizka, Mar 03 2009

a(n) is the number of unordered (not necessarily distinct) triples of subsets taken from the power set of {1,2,...,n}. Cf. A007582 (pairs of such subsets). - Geoffrey Critzer, Jul 10 2013

			a(5) = C(2+2^5,3) = C(34,3) = 5984.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

Cf. A000079, A007581, A092056, A016131.

[Binomial(2^n+2, 3): n in [0..30]]; // G. C. Greubel, Dec 27 2024

seq(binomial(2+2^n, 3), n=0..25); # Zerinvary Lajos, Feb 22 2008

nn=20;Table[Coefficient[Series[1/(1-x)^(2^n),{x,0,nn}],x^3],{n,0,nn}] (* Geoffrey Critzer, Jul 10 2013 *)
Binomial[2+2^Range[0,30], 3] (* G. C. Greubel, Dec 27 2024 *)

Vec((1-10*x+20*x^2)/((1-2*x)*(1-4*x)*(1-8*x)) + O(x^100)) \\ Colin Barker, Sep 13 2014

def A092055(n): return binomial(pow(2,n)+2,3)
print([A092055(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

a(n) = (2^(3*n-1) +3*2^(2*n-1) + 2^n)/3 = A092056(3, n).
a(n) = A000079(n)*A007581(n).
a(n) = 2*a(n-1) + 4^(n-1) + 8^(n-1).
a(n) = [x^3] 1/(1-x)^(2^n). - Geoffrey Critzer, Jul 11 2013
a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3). - Colin Barker, Sep 13 2014
G.f.: (1-10*x+20*x^2) / ((1-2*x)*(1-4*x)*(1-8*x)). - Colin Barker, Sep 13 2014
E.g.f.: (1/6)*(2*exp(2*x) + 3*exp(4*x) + exp(8*x)). - G. C. Greubel, Dec 27 2024

A383902 Square table read by ascending antidiagonals where T(n,k) = binomial(k+2^n-2,k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 28, 10, 1, 0, 1, 31, 120, 84, 15, 1, 0, 1, 63, 496, 680, 210, 21, 1, 0, 1, 127, 2016, 5456, 3060, 462, 28, 1, 0, 1, 255, 8128, 43680, 46376, 11628, 924, 36, 1, 0, 1, 511, 32640, 349504, 720720, 324632, 38760, 1716, 45, 1, 0

Offset: 0

Isaac R. Browne, May 15 2025

T(n,k) is the number of right total relations between a set of n distinguishable elements and a set of k indistinguishable elements.

			Rows start:
    1,  0,   0,   0,    0, ...
    1,  1,   1,   1,    1, ...
    1,  3,   6,  10,   15, ...
    1,  7,  28,  84,  210, ...
    1, 15, 120, 680, 3060, ...

Cf. A383905 (descending diagonals), A092056 (no restriction on totality)

T:= (n, k)-> binomial(k+2^n-2, k):
seq(seq(T(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, May 16 2025

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A060690 a(n) = binomial(2^n + n - 1, n).

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A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

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A092055 a(n) = binomial(2 + 2^n,3).

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A383902 Square table read by ascending antidiagonals where T(n,k) = binomial(k+2^n-2,k).

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Maple