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 1-8 of 8 results.

A117263 Row sums of triangle A117262; also, self-convolution of A117264.

Original entry on oeis.org

1, 2, 7, 64, 1729, 140050, 34032151, 24809438080, 54258241080961, 355988319732185122, 7006918097288599756327, 413751506726794527011353024, 73294838162131470076480154142529
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Crossrefs

Cf. A117261, A117262, A117264.

Programs

  • PARI
    a(n)=sum(k=0,n,3^((n-k)*(n+k-1)/2))

Formula

a(n) = Sum_{k=0..n} 3^( n*(n-1)/2 - k*(k-1)/2 ).
G.f. A(x) satisfies: A(x) = 1/(1 - x) + x * A(3*x). - Ilya Gutkovskiy, Jun 06 2020

A117260 Triangle T, read by rows, where matrix inverse T^-1 has -2^n in the secondary diagonal: [T^-1](n+1,n) = -2^n, with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 8, 8, 4, 1, 64, 64, 32, 8, 1, 1024, 1024, 512, 128, 16, 1, 32768, 32768, 16384, 4096, 512, 32, 1, 2097152, 2097152, 1048576, 262144, 32768, 2048, 64, 1, 268435456, 268435456, 134217728, 33554432, 4194304, 262144, 8192, 128, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=-1, q=2, r=1.
T(n,k) is the number of simple labeled graphs G on [n] such that the subgraph of G induced by the vertices labeled 1,2,...,k is a clique of size k. Cf A277219. - Geoffrey Critzer, May 05 2024

Examples

			Triangle T begins:
  1;
  1,1;
  2,2,1;
  8,8,4,1;
  64,64,32,8,1;
  1024,1024,512,128,16,1;
  32768,32768,16384,4096,512,32,1;
  2097152,2097152,1048576,262144,32768,2048,64,1;
  268435456,268435456,134217728,33554432,4194304,262144,8192,128,1;
Matrix inverse T^-1 has -2^n in the 2nd diagonal:
  1,
  -1,1,
  0,-2,1,
  0,0,-4,1,
  0,0,0,-8,1,
  0,0,0,0,-16,1,
  0,0,0,0,0,-32,1,
  ...

Crossrefs

Cf. A006125 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Programs

  • Maple
    T := (n, k) -> 2^(((n + k - 1)*(n - k))/2):
seq(seq(T(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Dec 31 2024
  • Mathematica
    Flatten[Table[2^((n(n-1))/2-(k(k-1))/2),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Sep 19 2013 *)
  • PARI
    {T(n,k)=local(m=1,p=-1,q=2,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Formula

T(n,k) = 2^(n*(n-1)/2 - k*(k-1)/2).

A117250 Triangle T, read by rows, where matrix power T^2 has powers of 2 in the secondary diagonal: [T^2](n+1,n) = 2^(n+1), with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 4, -4, 4, 1, -40, 32, -16, 8, 1, 896, -640, 256, -64, 16, 1, -43008, 28672, -10240, 2048, -256, 32, 1, 4325376, -2752512, 917504, -163840, 16384, -1024, 64, 1, -899678208, 553648128, -176160768, 29360128, -2621440, 131072, -4096, 128, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=2, q=2, r=1.

Examples

			Triangle T begins:
1;
1,1;
-1,2,1;
4,-4,4,1;
-40,32,-16,8,1;
896,-640,256,-64,16,1;
-43008,28672,-10240,2048,-256,32,1;
4325376,-2752512,917504,-163840,16384,-1024,64,1;
-899678208,553648128,-176160768,29360128,-2621440,131072,-4096,128,1;
Matrix square T^2 has powers of 2 in the 2nd diagonal:
1;
2,1;
0,4,1;
0,0,8,1;
0,0,0,16,1;
0,0,0,0,32,1;
0,0,0,0,0,64,1; ...

Crossrefs

Cf. A117251 (column 0); variants: A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Programs

  • PARI
    {T(n,k)=local(m=1,p=2,q=2,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Formula

T(n,k) = A117251(n-k)*2^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-2*j)]/(n-k)!*2^(n*(n-1)/2 - k*(k-1)/2) for n>k>=0, with T(n,n) = 1.

A117252 Triangle T, read by rows, where matrix power T^3 has powers of 3 in the secondary diagonal: [T^3](n+1,n) = 3^(n+1), with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, -3, 3, 1, 45, -27, 9, 1, -2430, 1215, -243, 27, 1, 433026, -196830, 32805, -2187, 81, 1, -245525742, 105225318, -15943230, 885735, -19683, 243, 1, 434685788658, -178988265918, 25569752274, -1291401630, 23914845, -177147, 729, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=3, q=3, r=1.

Examples

			Triangle T begins:
1;
1,1;
-3,3,1;
45,-27,9,1;
-2430,1215,-243,27,1;
433026,-196830,32805,-2187,81,1;
-245525742,105225318,-15943230,885735,-19683,243,1;
434685788658,-178988265918,25569752274,-1291401630,23914845,-177147,729,1;
Matrix cube T^3 has powers of 3 in the 2nd diagonal:
1;
3,1;
0,9,1;
0,0,27,1;
0,0,0,81,1;
0,0,0,0,243,1;
0,0,0,0,0,729,1; ...

Crossrefs

Cf. A117253 (column 0); variants: A117250 (p=q=2), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Programs

  • PARI
    {T(n,k)=local(m=1,p=3,q=3,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Formula

T(n,k) = A117253(n-k)*3^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-3*j)]/(n-k)!*3^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1.

A117254 Triangle T, read by rows, where matrix power T^4 has powers of 4 in the secondary diagonal: [T^4](n+1,n) = 4^(n+1), with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, -6, 4, 1, 224, -96, 16, 1, -39424, 14336, -1536, 64, 1, 30277632, -10092544, 917504, -24576, 256, 1, -98180268032, 31004295168, -2583691264, 58720256, -393216, 1024, 1, 1321338098679808, -402146377859072, 31748398252032, -661424963584, 3758096384, -6291456, 4096, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=4, q=4, r=1.

Examples

			Triangle T begins:
1;
1,1;
-6,4,1;
224,-96,16,1;
-39424,14336,-1536,64,1;
30277632,-10092544,917504,-24576,256,1;
-98180268032,31004295168,-2583691264,58720256,-393216,1024,1; ...
Matrix power T^4 has powers of 4 in the 2nd diagonal:
1;
4,1;
0,16,1;
0,0,64,1;
0,0,0,256,1;
0,0,0,0,1024,1;
0,0,0,0,0,4096,1; ...

Crossrefs

Cf. A117255 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Programs

  • PARI
    {T(n,k)=local(m=1,p=4,q=4,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Formula

T(n,k) = A117255(n-k)*4^((n-k)*k). T(n,k) = (-1)^(n-k)*4^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(4*j-1) for n>k>=0, with T(n,n) = 1.

A117256 Triangle T, read by rows, where matrix power T^5 has powers of 5 in the secondary diagonal: [T^5](n+1,n) = 5^(n+1), with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, -10, 5, 1, 750, -250, 25, 1, -328125, 93750, -6250, 125, 1, 779296875, -205078125, 11718750, -156250, 625, 1, -9741210937500, 2435302734375, -128173828125, 1464843750, -3906250, 3125, 1, 630569458007812500, -152206420898437500, 7610321044921875
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=5, q=5, r=1.

Examples

			Triangle T begins:
1;
1,1;
-10,5,1;
750,-250,25,1;
-328125,93750,-6250,125,1;
779296875,-205078125,11718750,-156250,625,1;
-9741210937500,2435302734375,-128173828125,1464843750,-3906250,3125,1;
Matrix power T^5 has powers of 5 in the 2nd diagonal:
1;
5,1;
0,25,1;
0,0,125,1;
0,0,0,625,1;
0,0,0,0,3125,1; ...

Crossrefs

Cf. A117257 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Programs

  • PARI
    {T(n,k)=local(m=1,p=5,q=5,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Formula

T(n,k) = A117257(n-k)*5^((n-k)*k). T(n,k) = (-1)^(n-k)*5^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(5*j-1) for n>k>=0, with T(n,n) = 1.

A117258 Triangle T, read by rows, where matrix power T^2 has 2*4^n in the secondary diagonal: [T^2](n+1,n) = 2*4^n, with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, -2, 4, 1, 32, -32, 16, 1, -2560, 2048, -512, 64, 1, 917504, -655360, 131072, -8192, 256, 1, -1409286144, 939524096, -167772160, 8388608, -131072, 1024, 1, 9070970929152, -5772436045824, 962072674304, -42949672960, 536870912, -2097152, 4096, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=2, q=4, r=1.

Examples

			Triangle T begins:
1;
1,1;
-2,4,1;
32,-32,16,1;
-2560,2048,-512,64,1;
917504,-655360,131072,-8192,256,1;
-1409286144,939524096,-167772160,8388608,-131072,1024,1;
Matrix square T^2 has 2*4^n in the 2nd diagonal:
1,
2,1,
0,8,1,
0,0,32,1,
0,0,0,128,1,
0,0,0,0,512,1, ...

Crossrefs

Cf. A117259 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (q=5), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Programs

  • PARI
    {T(n,k)=local(m=1,p=2,q=4,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Formula

T(n,k) = A117259(n-k)*4^((n-k)*k). T(n,k) = (-1)^(n-k)*4^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(2*j-1) for n>k>=0, with T(n,n) = 1.

A117265 Triangle T, read by rows, where matrix power T^-2 has -2^(n+1) in the secondary diagonal: [T^-2](n+1,n) = -2^(n+1), with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 20, 12, 4, 1, 280, 160, 48, 8, 1, 8064, 4480, 1280, 192, 16, 1, 473088, 258048, 71680, 10240, 768, 32, 1, 56229888, 30277632, 8257536, 1146880, 81920, 3072, 64, 1, 13495173120, 7197425664, 1937768448, 264241152, 18350080, 655360
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Comments

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=-2, q=2, r=1.

Examples

			Triangle T begins:
1;
1,1;
3,2,1;
20,12,4,1;
280,160,48,8,1;
8064,4480,1280,192,16,1;
473088,258048,71680,10240,768,32,1;
56229888,30277632,8257536,1146880,81920,3072,64,1;
13495173120,7197425664,1937768448,264241152,18350080,655360,12288,128,1;
Matrix inverse square T^-2 has -2^(n+1) in the 2nd diagonal:
1;
-2,1;
0,-4,1;
0,0,-8,1;
0,0,0,-16,1;
0,0,0,0,-32,1;
0,0,0,0,0,-64,1; ...

Crossrefs

Cf. A086229 (column 0), A117266 (row sums); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3).

Programs

  • PARI
    {T(n,k)=local(m=1,p=-2,q=2,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

Formula

T(n,k) = A086229(n-k)*2^((n-k)*k). T(n,k) = 2^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(2*j+1) for n>k>=0, with T(n,n) = 1.
Showing 1-8 of 8 results.

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