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.

A126461 Column 0 of triangle A126460; equals the number of subpartitions of the partition {(k^2 + 3*k - 4)*k/6, k>=0}.

Original entry on oeis.org

1, 1, 1, 3, 21, 274, 5806, 182766, 8034916, 471517614, 35682799508, 3388864405941, 395127873991296, 55543575452873070, 9271180003481197642, 1813921568747948684475, 411378931233397975750296
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Comments

When shifted left, equals column 1 of triangle A126460, which is the number of subpartitions of partition: {(k^2 + 6*k + 5)*k/6, k>=0}.

Examples

			Equals the number of subpartitions of the partition:
{(k^2 + 3*k - 4)*k/6, k>=0} = [0,0,2,7,16,30,50,77,112,156,210,275,...]
as illustrated by g.f.:
1/(1-x) = 1*(1-x)^0 + 1*x*(1-x)^0 + 1*x^2*(1-x)^2 + 3*x^3*(1-x)^7 + 21*x^4*(1-x)^16 + 274*x^5*(1-x)^30 + 5806*x^6*(1-x)^50 + 182766*x^7*(1-x)^77 ...

Crossrefs

Cf. A126460; A126462, A126463, A126464.

Programs

  • PARI
    {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(1+(k^2+3*k-4)*k/6)), n)}

Formula

G.f.: 1/(1-x) = Sum_{k>=0} a(k)*x^k*(1-x)^[(k^2 + 3*k - 4)*k/6].

A126462 Column 2 of triangle A126460; equals the number of subpartitions of the partition {(k^2 + 9*k + 20)*k/6, k>=0}.

Original entry on oeis.org

1, 1, 6, 75, 1565, 48950, 2145626, 125727238, 9507150815, 902519025315, 105203477607220, 14786330708536422, 2467862211341410635, 482812610434512386665, 109492763990117261581870
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Examples

			Equals the number of subpartitions of the partition:
{(k^2 + 9*k + 20)*k/6, k>=0} = [0,5,14,28,48,75,110,154,208,273,...]
as illustrated by g.f.:
1/(1-x) = 1*(1-x)^0 + 1*x*(1-x)^5 + 6*x^2*(1-x)^14 + 75*x^3*(1-x)^28 + 1565*x^4*(1-x)^48 + 48950*x^5*(1-x)^75 + 2145626*x^6*(1-x)^110 + 125727238*x^7*(1-x)^154 ...

Crossrefs

Cf. A126460; A126461, A126463, A126464.

Programs

  • PARI
    {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(1+(k^2+9*k+20)*k/6)), n)}

Formula

G.f.: 1/(1-x) = Sum_{k>=0} a(k)*x^k*(1-x)^[(k^2 + 9*k + 20)*k/6].

A126463 Column 3 of triangle A126460; equals the number of subpartitions of the partition {(k^2 + 9*k + 20)*k/6, k>=0}.

Original entry on oeis.org

1, 1, 10, 195, 5940, 257300, 14989472, 1130000385, 107089958760, 12470885416545, 1751753684302150, 292264756622072214, 57165584968923450000, 12962148519535236156640, 3374220800446022166695530
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Examples

			Equals the number of subpartitions of the partition:
{(k^2 + 12*k + 41)*k/6, k>=0} = [0,9,23,43,70,105,149,203,268,345,...]
as illustrated by g.f.:
1/(1-x) = 1*(1-x)^0 + 1*x*(1-x)^9 + 10*x^2*(1-x)^23 + 195*x^3*(1-x)^43 + 5940*x^4*(1-x)^70 + 257300*x^5*(1-x)^105 + 14989472*x^6*(1-x)^149 + 1130000385*x^7*(1-x)^203 ...

Crossrefs

Cf. A126460; A126461, A126462, A126464.

Programs

  • PARI
    {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(1+(k^2+12*k+41)*k/6)),n)}

Formula

G.f.: 1/(1-x) = Sum_{k>=0} a(k)*x^k*(1-x)^[(k^2 + 12*k + 41)*k/6].

A126464 Row sums of triangle A126460.

Original entry on oeis.org

1, 2, 3, 8, 50, 635, 13389, 420865, 18491156, 1084804118, 82081329459, 7794746829520, 908790397019076, 127745867968533747, 21322592031518420776, 4171751138526111626665, 946103460280012610769060
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2006

Keywords

Crossrefs

Cf. A126460; A126461, A126462, A126463.

Programs

  • PARI
    {a(n)=sum(k=0,n,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]))}

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.

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.

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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