A123306 -id:A123306 - cp's OEIS frontend

A123305 Triangle T, read by rows, where column k of T equals (k+1)(column k of T^2) when shifted to have an initial '1'; i.e., T(n,k) = (k+1)[T^2](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0.

1, 1, 1, 2, 2, 1, 6, 8, 3, 1, 26, 44, 18, 4, 1, 156, 312, 144, 32, 5, 1, 1234, 2776, 1422, 336, 50, 6, 1, 12340, 30312, 16848, 4256, 650, 72, 7, 1, 150994, 397728, 235458, 63072, 10050, 1116, 98, 8, 1, 2204112, 6151768, 3827628, 1076128, 178900, 20376, 1764

Offset: 0

Paul D. Hanna, Sep 24 2006

			Triangle begins:
  1;
  1, 1;
  2, 2, 1;
  6, 8, 3, 1;
  26, 44, 18, 4, 1;
  156, 312, 144, 32, 5, 1;
  1234, 2776, 1422, 336, 50, 6, 1;
  12340, 30312, 16848, 4256, 650, 72, 7, 1;
  150994, 397728, 235458, 63072, 10050, 1116, 98, 8, 1;
  ...
Matrix square starts:
  1;
  2, 1;
  6, 4, 1;
  26, 22, 6, 1;
  156, 156, 48, 8, 1;
  1234, 1388, 474, 84, 10, 1;
  12340, 15156, 5616, 1064, 130, 12, 1;
  ...
Note that (column k of T shifted) = (k+1)*(column k of T^2):
  k=1: [2,8,44,312,2776,...] = 2*[1,4,22,156,1388,...];
  k=2: [3,18,144,1422,16848,....] = 3*[1,6,48,474,5616,...].

Columns: A123306, A123307, A123308, A123309.

Cf. A123310 (row sums), A123311 (central terms).

{T(n, k)=if(n<0||k>n,0,if(n==k,1,(k+1)*sum(j=0, n-1, T(n-1, j)*T(j, k)); ))}

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

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

Paul D. Hanna, Sep 24 2006

nonn

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).

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

Paul D. Hanna, Sep 24 2006

nonn

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).

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

Paul D. Hanna, Sep 25 2006

nonn

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).

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

Paul D. Hanna, Sep 25 2006

nonn

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

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

Paul D. Hanna, Sep 25 2006

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.

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

Paul D. Hanna, Sep 25 2006

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.

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

cp's OEIS Frontend

A123305 Triangle T, read by rows, where column k of T equals (k+1)*(column k of T^2) when shifted to have an initial '1'; i.e., T(n,k) = (k+1)*[T^2](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0.

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A123307 Column 1 of triangle A123305.

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A123308 Column 2 of triangle A123305.

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A123309 Column 3 of triangle A123305.

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A123310 Row sums of triangle A123305.

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A123311 Central terms of triangle A123305.

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

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A123305 Triangle T, read by rows, where column k of T equals (k+1)(column k of T^2) when shifted to have an initial '1'; i.e., T(n,k) = (k+1)[T^2](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0.