A014070 -id:A014070 - cp's OEIS frontend

A136648 Inverse binomial transform of A014070: a(n) = Sum_{k=0..n} (-1)^(n-k)C(n,k)C(2^k,k).

1, 1, 3, 43, 1625, 192785, 73792371, 94005141667, 408909577044065, 6204433373664395569, 334203804752658372354515, 64828498485572980097719939179, 45811084061472137471487315433296153, 119028111984311982345314987179033877373025, 1145664208319965667452046935744516601565935434531

Offset: 0

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A014070 (C(2^n, n)), A134174.

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

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

/* Using the g.f.: */ {a(n)=my(X=x+x*O(x^n));polcoeff(sum(k=0,n,(log(1+(2^k+1)*X)-log(1+X))^k/k!)/(1+X),n)}

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

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

Edited by Charles R Greathouse IV, Oct 28 2009

Terms a(13) and beyond from Andrew Howroyd, Feb 02 2020

A136649 Binomial transform of A014070: a(n) = Sum_{k=0..n} C(n,k)*C(2^k,k).

Original entry on oeis.org

1, 3, 11, 81, 2089, 211107, 76211147, 95054910473, 410422012327681, 6211807332775516467, 334327967114349983723899, 64835852334793138873642165105, 45812640033676518721399820389451657

Offset: 0

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..59

Cf. A014070 (C(2^n, n)), A134173.

Partial sums of A180687.

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

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

/* Using the g.f.: */ {a(n)=local(X=x+x*O(x^n));polcoeff(sum(k=0,n,(log(1+(2^k-1)*X)-log(1-X))^k/k!)/(1-X),n)}

G.f.: A(x) = (1/(1-x))*Sum_{n>=0} [log(1 + (2^n-1)*x) - log(1-x)]^n / n!.
From Vaclav Kotesovec, Jul 02 2016: (Start)
a(n) ~ binomial(2^n,n).
a(n) ~ 2^(n^2) / n!.
a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)). (End)

Edited by Charles R Greathouse IV, Oct 28 2009

A140050 L.g.f.: A(x) = log(G(x)) where G(x) = g.f. of A014070(n) = C(2^n,n).

Original entry on oeis.org

2, 8, 140, 6840, 988272, 447403472, 660627632240, 3275795875152672, 55865676874553471264, 3342534504414732132234688, 713146571770256922167459951616, 549740788325926349073175414573702656

Offset: 1

Paul D. Hanna, May 02 2008

nonn

			A(x) = 2*x + 8*x^2/2 + 140*x^3/3 + 6840*x^4/4 + 988272*x^5/5 +...
A(x) = log(G(x)) where G(x) = g.f. of A014070:
G(x) = 1 + 2*x + 6*x^2 + 56*x^3 + 1820*x^4 +... + C(2^n,n)*x^n +...

Cf. A140051, A014070.

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

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

A136651 Self-convolution of A014070: a(n) = Sum_{k=0..n} C(2^k,k)*C(2^(n-k),n-k).

Original entry on oeis.org

1, 4, 16, 136, 3900, 410704, 150779216, 189354108224, 819706419291728, 12417873698752685696, 668556572391910046409088, 129665687275486846550512590336, 91623983383737723477835280780455168, 238057598315149125515904595621291745671168, 2291332225550784443587332334013451028612830795776

Offset: 0

Paul D. Hanna, Jan 16 2008

nonn

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

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

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

{a(n) = polcoeff( sum(m=0,n, sum(k=0,m, log(1+2^k*x +x*O(x^n))^k/k! * log(1+2^(m-k)*x +x*O(x^n))^(m-k) / (m-k)! ) ),n)}
for(n=0,20, print1(a(n),", "))

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

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

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

Original entry on oeis.org

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

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

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)

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

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

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A136648 Inverse binomial transform of A014070: a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*C(2^k,k).

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A136649 Binomial transform of A014070: a(n) = Sum_{k=0..n} C(n,k)*C(2^k,k).

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A140050 L.g.f.: A(x) = log(G(x)) where G(x) = g.f. of A014070(n) = C(2^n,n).

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A136651 Self-convolution of A014070: a(n) = Sum_{k=0..n} C(2^k,k)*C(2^(n-k),n-k).

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

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

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A136648 Inverse binomial transform of A014070: a(n) = Sum_{k=0..n} (-1)^(n-k)C(n,k)C(2^k,k).