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.

A123306 Column 0 of triangle A123305.

Original entry on oeis.org

1, 1, 2, 6, 26, 156, 1234, 12340, 150994, 2204112, 37617314, 738465620, 16451763834, 411281292920, 11427727238034, 350040470610996, 11736423864328466, 428080429306305136, 16893650334361392866
Offset: 0

Views

Author

Paul D. Hanna, Sep 24 2006

Keywords

Comments

Triangle T=A123305 is defined by: (column k of T) = (k+1)*(column k of T^2) shifted to have an initial '1'; thus this sequence shift left equals column 0 of T^2.

Crossrefs

Cf. A123305; A123307, A123308, A123309; A123310, A123311.

A123307 Column 1 of triangle A123305.

Original entry on oeis.org

1, 2, 8, 44, 312, 2776, 30312, 397728, 6151768, 110306160, 2260463464, 52304514912, 1352534122104, 38742215685776, 1219910431007656, 41945588389187712, 1565744883592488856, 63123204175617586000
Offset: 0

Views

Author

Paul D. Hanna, Sep 24 2006

Keywords

Comments

Triangle T=A123305 is defined by: (column k of T) = (k+1)*(column k of T^2) shifted to have an initial '1'; thus this sequence shifted left equals twice column 1 of T^2 (matrix square of A123305).

Crossrefs

Cf. A123305; A123306, A123308, A123309; A123310, A123311.

A123308 Column 2 of triangle A123305.

Original entry on oeis.org

1, 3, 18, 144, 1422, 16848, 235458, 3827628, 71466174, 1515031920, 36091609434, 957367079244, 28050237910926, 901359663633000, 31567805908329330, 1198307340146774844, 49060139971072916142, 2156816501291429473248
Offset: 0

Views

Author

Paul D. Hanna, Sep 24 2006

Keywords

Comments

Triangle T=A123305 is defined by: (column k of T) = (k+1)*(column k of T^2) shifted to have an initial '1'; thus this sequence shifted left equals 3 times column 2 of T^2 (matrix square of A123305).

Crossrefs

Cf. A123305, A123306, A123307, A123309, A123310, A123311.

A123309 Column 3 of triangle A123305.

Original entry on oeis.org

1, 4, 32, 336, 4256, 63072, 1076128, 20900032, 457645984, 11201485632, 304074996512, 9089787596288, 297305835048096, 10578368788224320, 407327197600166304, 16894650230806939776, 751638529171350907808
Offset: 0

Views

Author

Paul D. Hanna, Sep 25 2006

Keywords

Comments

Triangle T=A123305 is defined by: (column k of T) = (k+1)*(column k of T^2) shifted to have an initial '1'; thus this sequence shifted left equals 4 times column 3 of T^2 (matrix square of A123305).

Crossrefs

Cf. A123305; A123306, A123307, A123308; A123310, A123311.

A123310 Row sums of triangle A123305.

Original entry on oeis.org

1, 2, 5, 18, 93, 650, 5825, 64486, 858525, 13460814, 244360845, 5063982498, 118379934453, 3090176655290, 89294118056273, 2834776197278278, 98222473389128821, 3693056479700344798, 149906848766779722109
Offset: 0

Views

Author

Paul D. Hanna, Sep 25 2006

Keywords

Comments

Triangle T=A123305 is defined by: (column k of T) = (k+1)*(column k of T^2) shifted to have an initial '1'.

Crossrefs

Cf. A123305; A123306, A123307, A123308, A123309, A123311.

A123311 Central terms of triangle A123305.

Original entry on oeis.org

1, 2, 18, 336, 10050, 423576, 23317042, 1595336448, 131036683554, 12595263510000, 1389395816978554, 173214827643727872, 24105386508556401906, 3706800170875348334480, 624528505602985043491650
Offset: 0

Views

Author

Paul D. Hanna, Sep 25 2006

Keywords

Comments

Triangle T=A123305 is defined by: (column k of T) = (k+1)*(column k of T^2) shifted to have an initial '1'; thus A123312(n) = a(n)/(n+1) is an integer for n>=0.

Crossrefs

Cf. A123305; A123306, A123307, A123308, A123309, A123310, A123312.

A123312 a(n) = A123311(n)/(n+1), where A123311 forms the central terms of triangle A123305.

Original entry on oeis.org

1, 1, 6, 84, 2010, 70596, 3331006, 199417056, 14559631506, 1259526351000, 126308710634414, 14434568970310656, 1854260500658184762, 264771440776810595320, 41635233706865669566110
Offset: 0

Views

Author

Paul D. Hanna, Sep 25 2006

Keywords

Comments

Triangle T=A123305 is defined by: (column k of T) = (k+1)*(column k of T^2) shifted to have an initial '1'; thus a(n) = A123311(n)/(n+1) is an integer for n>=0.

Crossrefs

Cf. A123305; A123306, A123307, A123308, A123309, A123310, A123311.

A118024 Triangle T, read by rows, T(n,k) = T(n-k)*2^(k*(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)*2^(k*(n-k)) for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 8, 4, 1, 28, 48, 32, 8, 1, 216, 448, 384, 128, 16, 1, 2864, 6912, 7168, 3072, 512, 32, 1, 66656, 183296, 221184, 114688, 24576, 2048, 64, 1, 2760896, 8531968, 11730944, 7077888, 1835008, 196608, 8192, 128, 1, 205824384, 706789376
Offset: 0

Views

Author

Paul D. Hanna, Apr 10 2006

Keywords

Comments

Column 0 is A118025, where T(n,k) = A118025(n-k)*2^(k*(n-k)).

Examples

			Triangle T begins:
1;
1,1;
2,2,1;
6,8,4,1;
28,48,32,8,1;
216,448,384,128,16,1;
2864,6912,7168,3072,512,32,1;
66656,183296,221184,114688,24576,2048,64,1; ...
2760896,8531968,11730944,7077888,1835008,196608,8192,128,1; ...
Matrix square is given by [T^2](n,k) = T(n-k+1,0)*2^(k*(n-k)):
1;
2,1;
6,4,1;
28,24,8,1;
216,224,96,16,1;
2864,3456,1792,384,32,1; ...
so that column 0 of T^2 equals column 0 of T shift left 1 place.

Crossrefs

Cf. A118025 (column 0); A117401 (related triangle); A118022 (variant).
Cf. A123305.

Programs

  • PARI
    {T(n, k)=if(n<0 || k>n,0,if(n==k,1,2^k*sum(j=0, n-1, T(n-1, j)*T(j, k)); ))} \\ Paul D. Hanna, Sep 25 2006

Formula

T(n,k) = A118025(n-k)*2^(k*(n-k)) for n>=k>=0.
Showing 1-8 of 8 results.

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

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