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 1-7 of 7 results.

A126458 Column 0 of triangle A126457; a(n) = C( C(n+2,3) + 3, n).

Original entry on oeis.org

1, 4, 21, 286, 8855, 501942, 45057474, 5843355957, 1029873432159, 236236542585120, 68292983465630781, 24268885951464043344, 10392619362579990298763, 5276256293478688846049120
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126457, A126459; A126446, A126451, A126455.

Programs

  • PARI
    a(n)=binomial(n*(n+1)*(n+2)/3!+3, n)

A126459 Column 1 of triangle A126457; a(n) = C( C(n+3,3) + 2, n).

Original entry on oeis.org

1, 6, 66, 1540, 66045, 4582116, 470155077, 66983637864, 12655529067060, 3062465626261470, 923729223066105456, 339813167168828020668, 149762733221010818774320, 77904783726874238769542600
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126457, A126458; A126447, A126452, A126456.

Programs

  • PARI
    a(n)=binomial((n+1)*(n+2)*(n+3)/3!+2, n)

A126445 Triangle, read by rows, where T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) for n >= k >= 0.

Original entry on oeis.org

1, 1, 1, 6, 3, 1, 120, 36, 6, 1, 4845, 969, 120, 10, 1, 324632, 46376, 4495, 300, 15, 1, 32468436, 3478761, 270725, 15180, 630, 21, 1, 4529365776, 377447148, 24040016, 1150626, 41664, 1176, 28, 1, 840261910995, 56017460733, 2967205528, 122391522, 3921225, 98770, 2016, 36, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.

Examples

			Formula: T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) is illustrated by:
T(n=4,k=1) = C(C(6,3) - C(3,3), n-k) = C(19,3) = 969;
T(n=4,k=2) = C(C(6,3) - C(4,3), n-k) = C(16,2) = 120;
T(n=5,k=2) = C(C(7,3) - C(4,3), n-k) = C(31,3) = 4495.
Triangle begins:
           1;
           1,         1;
           6,         3,        1;
         120,        36,        6,       1;
        4845,       969,      120,      10,     1;
      324632,     46376,     4495,     300,    15,    1;
    32468436,   3478761,   270725,   15180,   630,   21,  1;
  4529365776, 377447148, 24040016, 1150626, 41664, 1176, 28, 1;

Links

Crossrefs

Columns: A126446, A126447, A126448, A126449 (row sums).
Variants: A107862, A126450, A126454, A126457.

Programs

  • Mathematica
    T[n_, k_]:= Binomial[Binomial[n+2,3] - Binomial[k+2,3], n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2022 *)
  • PARI
    T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!, n-k)
  • Sage
    def A126445(n,k): return binomial(binomial(n+2,3) - binomial(k+2,3), n-k)
flatten([[A126445(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 18 2022

Formula

T(n,k) = C(n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3!, n-k) for n >= k >= 0.

A126460 Triangle T, read by rows, where column k of matrix power T^( k(k+1)/2 ) equals left-shifted column (k-1) of T for k>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 21, 21, 6, 1, 1, 274, 274, 75, 10, 1, 1, 5806, 5806, 1565, 195, 15, 1, 1, 182766, 182766, 48950, 5940, 420, 21, 1, 1, 8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1, 471517614, 471517614, 125727238, 14989472, 1006880
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1. Also, column k equals unsigned column k of the matrix inverse of triangle P_k defined by P_k(m,j) = C( C(j+2,3) - C(k+2,3) + m-j, m-j) for m>=j>=0.

Examples

			Triangle T begins:
1;
1, 1;
1, 1, 1;
3, 3, 1, 1;
21, 21, 6, 1, 1;
274, 274, 75, 10, 1, 1;
5806, 5806, 1565, 195, 15, 1, 1;
182766, 182766, 48950, 5940, 420, 21, 1, 1;
8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1; ...
where column 1 of T^1 equals left-shifted column 0 of T.
Matrix cube T^3 begins:
1;
3, 1;
6, 3, (1);
22, 12, (3), 1;
163, 91, (21), 3, 1;
2167, 1219, (274), 33, 3, 1;
46248, 26091, (5806), 661, 48, 3, 1;
1460301, 824853, (182766), 20341, 1369, 66, 3, 1; ...
where column 2 of T^3 equals left-shifted column 1 of T.
Matrix power T^6 begins:
1;
6, 1;
21, 6, 1;
98, 33, 6, (1);
791, 281, 51, (6), 1;
10850, 3929, 710, (75), 6, 1;
234472, 85557, 15425, (1565), 105, 6, 1;
7444172, 2725402, 490806, (48950), 3080, 141, 6, 1; ...
where column 3 of T^6 equals left-shifted column 2 of T.

Crossrefs

Columns: A126461, A126462, A126463, A126464; A126465 (dual); A107876 (variant); subpartitions defined: A115728.

Programs

  • PARI
    {T(n,k)=abs((matrix(n+1,n+1,r,c, binomial((c-1)*c*(c+1)/3!-k*(k+1)*(k+2)/3!+r-c,r-c))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
  • PARI
    /* As Defined by Matrix Product A126460 = A126445^-1*A126450: */
{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!,r-c))), N=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1,r-c)))); (M^-1*N)[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

G.f. of column k: 1/(1-x) = Sum_{n>=0} T(n+k,k)*x^n*(1-x)^p_k(n), so that column k equals the number of subpartitions of the partition p_k defined by: p_k(n) = (n^2 + (3*k+3)*n + (3*k^2+6*k-4))*n/6 for n>=0.

A126465 Triangle T, read by rows, where row n equals row (n-1) of matrix power T^(n(n+1)/2) concatenated with a trailing '1', for n>0, with T(0,0) = 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 33, 6, 1, 1, 855, 105, 10, 1, 1, 40475, 3710, 255, 15, 1, 1, 3039204, 219625, 11935, 525, 21, 1, 1, 331630320, 19545316, 879571, 31584, 966, 28, 1, 1, 49563943161, 2437990653, 93365328, 2856819, 72786, 1638, 36, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

Amazingly, A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1; and also A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457.

Examples

			Triangle T begins:
1,
1, 1,
3, 1, 1,
33, 6, 1, 1,
855, 105, 10, 1, 1,
40475, 3710, 255, 15, 1, 1,
3039204, 219625, 11935, 525, 21, 1, 1,
331630320, 19545316, 879571, 31584, 966, 28, 1, 1,
49563943161, 2437990653, 93365328, 2856819, 72786, 1638, 36, 1, 1, ...
Matrix cube T^3 begins:
1;
[3, 1]; <-- row 1 of T^3 + '1' = row 2 of T;
12, 3, 1; ...
Matrix power T^6 begins:
1;
6, 1;
[33, 6, 1]; <-- row 2 of T^6 + '1' = row 3 of T.
Matrix power T^10 begins:
1;
10, 1;
75, 10, 1;
[855, 105, 10, 1]; <-- row 3 of T^10 + '1' = row 4 of T.
Matrix power T^15 begins:
1;
15, 1;
150, 15, 1;
1895, 195, 15, 1;
[40475, 3710, 255, 15, 1]; <-- row 4 of T^15 + '1' = row 5 of T.

Crossrefs

Columns: A126466, A126467, A126468; A126469 (row sums); A126460 (dual); A101479 (variant).

Programs

  • PARI
    {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!,r-c))), N=matrix(n+1,n+1,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1,r-c)))); (N*M^-1)[n+1,k+1]}

A126450 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 1, n-k) for n>=k>=0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 165, 45, 7, 1, 5985, 1140, 136, 11, 1, 376992, 52360, 4960, 325, 16, 1, 36288252, 3819816, 292825, 16215, 666, 22, 1, 4935847320, 406481544, 25621596, 1215450, 43680, 1225, 29, 1, 899749479915, 59487568920, 3127595016, 128164707
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.

Examples

			Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 1, n-k) is illustrated by:
T(n=4,k=1) = C( C(6,3) - C(3,3) + 1, n-k) = C(20,3) = 1140;
T(n=4,k=2) = C( C(6,3) - C(4,3) + 1, n-k) = C(17,2) = 136;
T(n=5,k=2) = C( C(7,3) - C(4,3) + 1, n-k) = C(32,3) = 4960.
Triangle begins:
1;
2, 1;
10, 4, 1;
165, 45, 7, 1;
5985, 1140, 136, 11, 1;
376992, 52360, 4960, 325, 16, 1;
36288252, 3819816, 292825, 16215, 666, 22, 1;
4935847320, 406481544, 25621596, 1215450, 43680, 1225, 29, 1; ...

Crossrefs

Columns: A126451, A126452; A126453 (row sums); variants: A126445, A126454, A126457, A107867.

Programs

  • PARI
    T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+1, n-k)

Formula

T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 1, n-k) for n>=k>=0.

A126454 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) for n>=k>=0.

Original entry on oeis.org

1, 3, 1, 15, 5, 1, 220, 55, 8, 1, 7315, 1330, 153, 12, 1, 435897, 58905, 5456, 351, 17, 1, 40475358, 4187106, 316251, 17296, 703, 23, 1, 5373200880, 437353560, 27285336, 1282975, 45760, 1275, 30, 1, 962889794295, 63140314380, 3295144749, 134153712
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.

Examples

			Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) is illustrated by:
T(n=4,k=1) = C( C(6,3) - C(3,3) + 2, n-k) = C(21,3) = 1330;
T(n=4,k=2) = C( C(6,3) - C(4,3) + 2, n-k) = C(18,2) = 153;
T(n=5,k=2) = C( C(7,3) - C(4,3) + 2, n-k) = C(33,3) = 5456.
Triangle begins:
1;
3, 1;
15, 5, 1;
220, 55, 8, 1;
7315, 1330, 153, 12, 1;
435897, 58905, 5456, 351, 17, 1;
40475358, 4187106, 316251, 17296, 703, 23, 1;
5373200880, 437353560, 27285336, 1282975, 45760, 1275, 30, 1; ...

Crossrefs

Columns: A126455, A126456; variants: A126445, A126450, A126457, A107870.

Programs

  • Mathematica
    Table[Binomial[Binomial[n+2,3]-Binomial[k+2,3]+2,n-k],{n,0,10},{k,0,n}]// Flatten (* Harvey P. Dale, Dec 17 2020 *)
  • PARI
    T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+2, n-k)

Formula

T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 2, n-k) for n>=k>=0.
Showing 1-7 of 7 results.

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