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 1-10 of 12 results. Next

A050158 T(n, k) = S(2*n + 1, n, k + 1) for 0<=k<=n and n >= 0, array S as in A050157.

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 14, 28, 34, 35, 42, 90, 117, 125, 126, 132, 297, 407, 451, 461, 462, 429, 1001, 1430, 1638, 1703, 1715, 1716, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435, 4862, 11934, 18122, 21930, 23630, 24174, 24293
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Triangle starts:
                               1
                              2, 3
                            5, 9, 10
                         14, 28, 34, 35
                     42, 90, 117, 125, 126
                  132, 297, 407, 451, 461, 462
            429, 1001, 1430, 1638, 1703, 1715, 1716

Crossrefs

T(n, 0) = A000108(n+1).
T(n, 1) = A000245(n+1).
T(n, n) = A001700(n).
T(n,n-1) = A010763(n).
Row sums are A296770.
Cf. A039598, A050157.

Programs

  • Maple
    A050158 := (n, k) ->  binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1):
seq(seq(A050158(n,k), k=0..n), n=0..6); # Peter Luschny, Dec 22 2017

Formula

T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A039598.
T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1). Peter Luschny, Dec 22 2017

A050159 T(n,k) = S(2n-1,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 5, 9, 10, 5, 14, 28, 34, 35, 14, 42, 90, 117, 125, 126, 42, 132, 297, 407, 451, 461, 462, 132, 429, 1001, 1430, 1638, 1703, 1715, 1716, 429, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435, 1430, 4862, 11934, 18122
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Rows: {0}; {1,1}; {1,2,3}; ...

Crossrefs

Cf. A050144, A050157.

Formula

T(n, k) = Sum_{t(n, j): 0<=j<=k}, array t as in A050144.

A050163 T(n, k) = S(2n+2, n+2, k+2) for 0<=k<=n and n >= 0, array S as in A050157.

Original entry on oeis.org

1, 3, 4, 9, 14, 15, 28, 48, 55, 56, 90, 165, 200, 209, 210, 297, 572, 726, 780, 791, 792, 1001, 2002, 2639, 2912, 2989, 3002, 3003, 3432, 7072, 9620, 10880, 11320, 11424, 11439, 11440, 11934, 25194, 35190, 40698, 42942, 43605
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Triangle starts:
                                1
                              3, 4
                           9, 14, 15
                         28, 48, 55, 56
                     90, 165, 200, 209, 210
                  297, 572, 726, 780, 791, 792
            1001, 2002, 2639, 2912, 2989, 3002, 3003

Crossrefs

T(n, 0) = A000245(n+1).
T(n, 1) = A002057(n).
T(n, n) = A001791(n+1).
Row sums are A000531(n+1).
Cf. A050155, A050157.

Programs

  • Maple
    A050163 := (n, k) -> binomial(2*n+2, n) - binomial(2*n+2, n+k+3):
seq(seq(A050163(n,k), k=0..n), n=0..8); # Peter Luschny, Dec 21 2017

Formula

T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A050155.
T(n, k) = binomial(2*n+2, n) - binomial(2*n+2, n+k+3). - Peter Luschny, Dec 21 2017

A296771 Row sums of A050157.

Original entry on oeis.org

1, 3, 13, 58, 257, 1126, 4882, 20980, 89497, 379438, 1600406, 6720748, 28117498, 117254268, 487589572, 2022568168, 8371423177, 34581780478, 142605399982, 587138954428, 2413944555742, 9911778919348, 40650232625212, 166534680737368, 681576405563722
Offset: 0

Views

Author

Peter Luschny, Dec 21 2017

Keywords

Links

Crossrefs

Cf. A000346, A037965, A050157, A296770.

Programs

  • Maple
    A296771 := n -> add(binomial(2*n, n) - binomial(2*n, n+k+1), k=0..n):
seq(A296771(n), n=0..24);
  • Mathematica
    a[n_] := 4^n ((n - 1/2)! (2 n + 3)/(2 Sqrt[Pi] n!) - 1/2);
Table[a[n], {n, 0, 24}]
  • PARI
    a(n) = sum(k=0, n, binomial(2*n, n) - binomial(2*n, n+k+1)) \\ Iain Fox, Dec 21 2017

Formula

a(n) = Sum_{k=0..n} (binomial(2*n, n) - binomial(2*n, n+k+1)).
a(n) = 2^(2*n-1)*(((n-1/2)!*(2*n+3))/(sqrt(Pi)*n!) - 1).
a(n) ~ 4^n*(sqrt(n/Pi) - 1/2).
a(n) = A037965(n+1) - A000346(n-1) for n >= 1.
From Robert Israel, Dec 21 2017: (Start)
a(n) = (n+3/2)*binomial(2*n,n) - 2^(2*n-1).
G.f.: (3/2-4*x)*(1-4*x)^(-3/2) - (1/2)*(1-4*x)^(-1).
64*(n+1)*(2*n+1)*a(n)-8*(2*n+3)*(5*n+4)*a(n+1)+2*(n+2)*(8*n+11)*a(n+2)-(n+3)*(n+2)*a(n+3)=0. (End)

A050160 T(n,k) = S(2n,n-1,k-1), 0 <= k <= n, n >= 0, array S as in A050157.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 9, 14, 15, 14, 28, 48, 55, 56, 42, 90, 165, 200, 209, 210, 132, 297, 572, 726, 780, 791, 792, 429, 1001, 2002, 2639, 2912, 2989, 3002, 3003, 1430, 3432, 7072, 9620, 10880, 11320, 11424, 11439, 11440, 4862, 11934, 25194, 35190, 40698, 42942
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Rows: {0}; {1,1}; {2,3,4}; ...

Formula

T(n, k) = Sum_{j=0..k} t(n, j), array t as in A050145.

Extensions

a(9) corrected and more terms from Sean A. Irvine, Aug 08 2021

A050161 T(n,k)=S(2n+1,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 9, 14, 20, 21, 28, 48, 75, 83, 84, 90, 165, 275, 319, 329, 330, 297, 572, 1001, 1209, 1274, 1286, 1287, 1001, 2002, 3640, 4550, 4900, 4990, 5004, 5005, 3432, 7072, 13260, 17068, 18768, 19312, 19431, 19447
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Rows: {0}; {1,1}; {3,4,5}; ...

Formula

T(n, k)=Sum{t(n, j): 0<=j<=k}, array t as in A050153.

A050162 T(n,k)=S(2n+2,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

Original entry on oeis.org

0, 1, 1, 4, 5, 6, 14, 20, 27, 28, 48, 75, 110, 119, 120, 165, 275, 429, 483, 494, 495, 572, 1001, 1638, 1911, 1988, 2001, 2002, 2002, 3640, 6188, 7448, 7888, 7992, 8007, 8008, 7072, 13260, 23256, 28764, 31008, 31671, 31806, 31823
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Rows: {0}; {1,1}; {4,5,6}; ...

Formula

T(n, k)=Sum{t(n, j): 0<=j<=k}, array t as in A050154.

A050164 T(n,k)=S(2n+3,n+3,k+3), 0<=k<=n, n >= 0, array S as in A050157.

Original entry on oeis.org

1, 4, 5, 14, 20, 21, 48, 75, 83, 84, 165, 275, 319, 329, 330, 572, 1001, 1209, 1274, 1286, 1287, 2002, 3640, 4550, 4900, 4990, 5004, 5005, 7072, 13260, 17068, 18768, 19312, 19431, 19447, 19448
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Rows: {1}; {4,5}; {14,20,21}; ...

Formula

T(n, k)=Sum{t(n, j): 0<=j<=k}, array t as in A050156.

A050174 T(n,k) = S(n,k,k-2), 1<=k<=n-2, n >= 3, array S as in A050157.

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 4, 9, 14, 9, 5, 14, 28, 28, 14, 6, 20, 48, 62, 48, 20, 7, 27, 75, 117, 117, 75, 27, 8, 35, 110, 200, 242, 200, 110, 35, 9, 44, 154, 319, 451, 451, 319, 154, 44, 10, 54, 208, 483, 780, 912, 780, 483, 208, 54, 11, 65, 273, 702, 1274
Offset: 3

Views

Author

Clark Kimberling

Keywords

Comments

S(n,k,k) = C(n,k) for 1<=k<=n, n >= 1; cf. Pascal's triangle, A007318.
S(n,k,k-1), 1<=k<=n-1, n >= 2 is given by A014430.

Examples

			Rows:
1;
2, 2;
3, 5, 5;
4, 9, 14, 9;
5, 14, 28, 28, 14;
...

Crossrefs

Cf. A007318, A014430.

Formula

T(n,k) = C(n,k) - n for k>1, and T(n, 1) = n - 2. - Andrei Asinowski, Jan 27 2016

Extensions

Offset changed to 3 by Michel Marcus, Jan 29 2016

A050175 T(n,k)=S(n,k,k-3), 2<=k<=n-3, n >= 5, array S as in A050157.

Original entry on oeis.org

2, 5, 5, 9, 14, 14, 14, 28, 42, 28, 20, 48, 90, 90, 48, 27, 75, 165, 207, 165, 75, 35, 110, 275, 407, 407, 275, 110, 44, 154, 429, 726, 858, 726, 429, 154, 54, 208, 637, 1209, 1638, 1638, 1209, 637, 208, 65, 273, 910, 1911, 2912, 3341
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			Rows: {2}; {5,5}; {9,14,14}; ...
Showing 1-10 of 12 results. Next

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

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