cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A137156 Matrix inverse of triangle A137153(n,k) = C(2^k+n-k-1, n-k), read by rows.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 9, -24, 22, -8, 1, -88, 239, -228, 92, -16, 1, 1802, -4920, 4749, -1976, 376, -32, 1, -75598, 206727, -200240, 84086, -16432, 1520, -64, 1, 6421599, -17568408, 17034964, -7173240, 1413084, -133984, 6112, -128, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2008

Keywords

Comments

Unsigned column 0 = A001192, number of full sets of size n.

Examples

			Triangle begins:
        1;
       -1,         1;
        1,        -2,        1;
       -2,         5,       -4,        1;
        9,       -24,       22,       -8,       1;
      -88,       239,     -228,       92,     -16,       1;
     1802,     -4920,     4749,    -1976,     376,     -32,    1;
   -75598,    206727,  -200240,    84086,  -16432,    1520,  -64,    1;
  6421599, -17568408, 17034964, -7173240, 1413084, -133984, 6112, -128, 1;
  ...

Crossrefs

Cf. A137153 (matrix inverse); unsigned columns: A001192, A137157, A137158, A137159; unsigned row sums: A137160.

Programs

  • PARI
    /* As matrix inverse of A137153: */
{T(n,k) = local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(2^(c-1)+r-c-1,r-c)))); if(n
  • PARI
    /* Using the g.f.: */
{T(n,k) = if(n

Formula

G.f. of column k: 1 = Sum_{n>=0} T(n+k,k)*x^n/(1-x)^(2^(n+k)).

A137154 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, n-k); equals the row sums of triangle A137153.

Original entry on oeis.org

1, 2, 4, 9, 24, 79, 331, 1803, 12954, 123983, 1592513, 27604172, 648528166, 20722205191, 903019659239, 53792176322629, 4388683843024734, 491232972054490915, 75545748143323475653, 15984344095578889888206
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2008

Keywords

Comments

Matrix inverse of A137153 is A137156.

Crossrefs

Cf. A137153, A137155, A137156.

Programs

  • Mathematica
    Table[Sum[Binomial[2^(n-k) + k - 1, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2021 *)
  • PARI
    a(n)=sum(k=0,n,binomial(2^k+n-k-1,n-k))
  • PARI
    {a(n)=local(A=sum(k=0,n,x^k/(1-x+x*O(x^n))^(2^k)));polcoeff(A,n)} \\ Paul D. Hanna, Sep 15 2009

Formula

G.f.: Sum_{n>=0} x^n/(1-x)^(2^n). - Paul D. Hanna, Sep 15 2009
G.f.: Sum_{n>=0} ( (-log(1 - x))^n / n! ) / (1 - 2^n*x). - Paul D. Hanna, Jan 23 2021

A137155 a(n) = Sum_{k=0..[n/2]} C(2^k + n-2k-1, n-2k); equals the antidiagonal sums of triangle A137153.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 17, 35, 80, 201, 561, 1771, 6298, 25217, 115440, 605243, 3609754, 24742305, 196470603, 1792166979, 18817616610, 230233601737, 3268914688382, 53509699527491, 1020486454701731, 22777217813040393, 589097686208575727
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2008

Keywords

Crossrefs

Cf. A137153, A137154.

Programs

  • PARI
    a(n)=sum(k=0,n\2,binomial(2^k+n-2*k-1,n-2*k))

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Magma
    [Binomial(2^n +n-1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
  • Maple
    with(combinat): for n from 0 to 20 do printf(`%d,`,binomial(2^n+n-1, n)) od:
  • Mathematica
    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)
  • PARI
    {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
  • PARI
    a(n) = sum(k=0, n, stirling(n,k,1)*(2^n+n-1)^k/n!); \\ Paul D. Hanna, Nov 20 2014
  • Python
    from math import comb
def A060690(n): return comb((1<Chai Wah Wu, Jul 05 2024
  • Sage
    [binomial(2^n +n-1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

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

Extensions

More terms from James Sellers, Apr 20 2001
Edited by N. J. A. Sloane, Mar 17 2008

A092056 Square table read by downward antidiagonals where T(n,k) = binomial(n+2^k-1,n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 10, 4, 1, 1, 16, 36, 20, 5, 1, 1, 32, 136, 120, 35, 6, 1, 1, 64, 528, 816, 330, 56, 7, 1, 1, 128, 2080, 5984, 3876, 792, 84, 8, 1, 1, 256, 8256, 45760, 52360, 15504, 1716, 120, 9, 1, 1, 512, 32896, 357760, 766480, 376992, 54264, 3432, 165, 10, 1
Offset: 0

Views

Author

Henry Bottomley, Feb 19 2004

Keywords

Comments

Each column is convolution of preceding column starting from the all 1's sequence.
T(n,k) is the number of relations between a set of k distinguishable elements and a set of n indistinguishable elements. - Isaac R. Browne, May 14 2025

Examples

			Rows start:
  1, 1,  1,   1,    1,     1,      1,...
  1, 2,  4,   8,   16,    32,     64,...
  1, 3, 10,  36,  136,   528,   2080,...
  1, 4, 20, 120,  816,  5984,  45760,...
  1, 5, 35, 330, 3876, 52360, 766480,...
  ...

Crossrefs

Columns include (essentially) A000012, A000027, A000292, A000580, A010968, etc.
Rows include A000012, A000079, A007582, A092056.
Main diagonal gives A060690.
Cf. A137153 (same with reflected antidiagonals).

Formula

T(n,k) = Sum_{i=0..n} T(i,k-1)*T(n-i,k-1) starting with T(n,0) = 1 for n>=0.

A383905 Square table read by descending antidiagonals where T(n,k) = binomial(k+2^n-2,k).

Original entry on oeis.org

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

Views

Author

Isaac R. Browne, May 15 2025

Keywords

Comments

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

Examples

			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, ...

Crossrefs

Cf. A383902 (ascending diagonals), A137153 (no restriction on totality).

Programs

  • Maple
    T:= (n, k)-> binomial(k+2^n-2, k):
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, May 16 2025
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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