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 1-9 of 9 results.

A117261 Row sums of triangle A117260.

Original entry on oeis.org

1, 2, 5, 21, 169, 2705, 86561, 5539905, 709107841, 181531607297, 92944182936065, 95174843326530561, 194918079132734588929, 798384452127680876253185, 6540365431829961738266091521, 107157347235102093119751643480065, 3511331954199825387348021853554769921
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2006

Keywords

Crossrefs

Cf. A073587, A117260, A117263.

Programs

  • Mathematica
    Table[Sum[2^((n(n-1))/2-(k(k-1))/2),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 05 2023 *)
  • PARI
    a(n)=sum(k=0,n,2^((n-k)*(n+k-1)/2))

Formula

a(n) = Sum_{k=0..n} 2^(n*(n-1)/2 - k*(k-1)/2).
G.f. A(x) satisfies: A(x) = 1/(1 - x) + x * A(2*x). - Ilya Gutkovskiy, Jun 06 2020
a(n) = a(n-1) * 2^(n-1) + 1 for n > 0 and a(0) = 1. - Werner Schulte, Oct 17 2023

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.

A117262 Triangle T, read by rows, where matrix inverse T^-1 has -3^n in the secondary diagonal: [T^-1](n+1,n) = -3^n, with all 1's in the main diagonal and zeros elsewhere.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 27, 27, 9, 1, 729, 729, 243, 27, 1, 59049, 59049, 19683, 2187, 81, 1, 14348907, 14348907, 4782969, 531441, 19683, 243, 1, 10460353203, 10460353203, 3486784401, 387420489, 14348907, 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=-1, q=3, r=1.

Examples

			Triangle T begins:
1;
1,1;
3,3,1;
27,27,9,1;
729,729,243,27,1;
59049,59049,19683,2187,81,1;
14348907,14348907,4782969,531441,19683,243,1;
10460353203,10460353203,3486784401,387420489,14348907,177147,729,1;
Matrix inverse T^-1 has -3^n in the 2nd diagonal:
1,
-1,1,
0,-3,1,
0,0,-9,1,
0,0,0,-27,1,
0,0,0,0,-81,1,
0,0,0,0,0,-243,1, ...

Crossrefs

Cf. A047656 (column 0), A117263 (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), A117265 (p=-2, q=2).

Programs

  • PARI
    {T(n,k)=local(m=1,p=-1,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) = 3^(n*(n-1)/2 - k*(k-1)/2).

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.

A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 8, 24, 24, 6, 64, 256, 384, 192, 24, 1024, 5120, 10240, 7680, 1920, 120, 32768, 196608, 491520, 491520, 184320, 23040, 720, 2097152, 14680064, 44040192, 55050240, 27525120, 5160960, 322560, 5040, 268435456, 2147483648, 7516192768, 11274289152, 7046430720, 1761607680, 165150720, 5160960, 40320
Offset: 0

Views

Author

Thomas Scheuerle, Sep 14 2023

Keywords

Comments

A tournament is a directed digraph obtained by assigning a direction for each edge in an undirected complete graph. In a transitive tournament all nodes can be strictly ordered by their reachability.

Examples

			Triangle begins:
     1
     1,     1
     2,     4,     2
     8,    24,    24,     6
    64,   256,   384,   192,    24
  1024,  5120, 10240,  7680,  1920,  120

Links

Crossrefs

Cf. A002866, A006125, A052670, A095340, A103904, A008279, A117260, A379614 (row sums).
Cf. A122027, A224886, A259105, A350608, A350609, A350610.

Programs

  • Maple
    T := (n, k) -> 2^(((n-1)*n - (k-1)*k)/2) * n! / (n-k)!:
seq(seq(T(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Nov 02 2023
  • PARI
    T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2))

Formula

T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2)).
T(n, 0) = A006125(n).
T(n, 1) = A095340(n).
T(n, 2) = A103904(n).
T(n, n) = n!.
T(n, n-1) = A002866(n-1).
T(n, n-2) = A052670(n).
T(n, k) = A008279(n, k) * A117260(n, k). - Peter Luschny, Dec 31 2024
Showing 1-9 of 9 results.

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