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.

A126466 Column 0 of triangle A126465.

Original entry on oeis.org

1, 1, 3, 33, 855, 40475, 3039204, 331630320, 49563943161, 9722245517970, 2423526680920125, 748560249534719259, 280702726068079355157, 125661789748507921799800, 66215986172320048509110050
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126465, A126467, A126468; A126469 (row sums).

Programs

  • PARI
    {a(n)=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+1,1]}

A126467 Column 1 of triangle A126465.

Original entry on oeis.org

1, 1, 6, 105, 3710, 219625, 19545316, 2437990653, 405899413380, 87002614878795, 23351252019790310, 7676367378822833691, 3035533054438876836390, 1422385798272704860377055, 779768824083518449340320720
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126465, A126466, A126468; A126469 (row sums).

Programs

  • PARI
    {a(n)=local(M=matrix(n+2,n+2,r,c,if(r>=c,binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!,r-c))), N=matrix(n+2,n+2,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+2,2]}

A126468 Column 2 of triangle A126465.

Original entry on oeis.org

1, 1, 10, 255, 11935, 879571, 93365328, 13464554610, 2531141851260, 601339958510170, 176255315943299736, 62514948119465302825, 26407795709641456484010, 13109371824722841349352160
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126465, A126466, A126467; A126469 (row sums).

Programs

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

A126469 Row sums of triangle A126465.

Original entry on oeis.org

1, 2, 5, 41, 972, 44457, 3271312, 352087787, 52098230423, 10141977387056, 2513121624034518, 772525844883337679, 288558785171538716889, 128760943237115947732171, 67665205307818875323789155
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126465, A126466, A126467, A126468.

Programs

  • PARI
    {a(n)=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)))); sum(k=0,n,(N*M^-1)[n+1,k+1])}

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.

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.

A126457 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 3, n-k) for n>=k>=0.

Original entry on oeis.org

1, 4, 1, 21, 6, 1, 286, 66, 9, 1, 8855, 1540, 171, 13, 1, 501942, 66045, 5984, 378, 18, 1, 45057474, 4582116, 341055, 18424, 741, 24, 1, 5843355957, 470155077, 29034396, 1353275, 47905, 1326, 31, 1, 1029873432159, 66983637864, 3470108187, 140364532
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) + 3, n-k) is illustrated by:
T(n=4,k=1) = C( C(6,3) - C(3,3) + 3, n-k) = C(22,3) = 1540;
T(n=4,k=2) = C( C(6,3) - C(4,3) + 3, n-k) = C(19,2) = 171;
T(n=5,k=2) = C( C(7,3) - C(4,3) + 3, n-k) = C(34,3) = 5984.
Triangle begins:
1;
4, 1;
21, 6, 1;
286, 66, 9, 1;
8855, 1540, 171, 13, 1;
501942, 66045, 5984, 378, 18, 1;
45057474, 4582116, 341055, 18424, 741, 24, 1;
5843355957, 470155077, 29034396, 1353275, 47905, 1326, 31, 1; ...

Crossrefs

Columns: A126458, A126459; variants: A126445, A126450, A126454, A107873.

Programs

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

Formula

T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 3, n-k) for n>=k>=0.
Showing 1-9 of 9 results.

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