A107863 -id:A107863 - cp's OEIS frontend

A107862 Triangle, read by rows, where T(n,k) = C(n(n-1)/2 - k(k-1)/2 + n-k, n-k).

1, 1, 1, 3, 2, 1, 20, 10, 3, 1, 210, 84, 21, 4, 1, 3003, 1001, 220, 36, 5, 1, 54264, 15504, 3060, 455, 55, 6, 1, 1184040, 296010, 53130, 7315, 816, 78, 7, 1, 30260340, 6724520, 1107568, 142506, 14950, 1330, 105, 8, 1, 886163135, 177232627, 26978328, 3262623, 324632, 27405, 2024, 136, 9, 1

Offset: 0

Paul D. Hanna, Jun 04 2005

Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.

			Triangle begins:
        1;
        1,      1;
        3,      2,     1;
       20,     10,     3,    1;
      210,     84,    21,    4,   1;
     3003,   1001,   220,   36,   5,  1;
    54264,  15504,  3060,  455,  55,  6, 1;
  1184040, 296010, 53130, 7315, 816, 78, 7, 1; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A014068 (column 0), A107863 (column 1), A099121 (column 2), A107865, A107867, A107870, A107876.

[Binomial(Floor((n-k)*(n+k+1)/2), n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2022

T[n_,k_]:= Binomial[(n-k)*(n+k+1)/2, n-k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 19 2022 *)

T(n,k)=binomial(n*(n-1)/2-k*(k-1)/2+n-k,n-k)

flatten([[binomial( (n-k)*(n+k+1)/2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2022

T(n,k) = binomial( (n-k)*(n+k+1)/2, n-k). - G. C. Greubel, Feb 19 2022

A135861 a(n) = binomial(n*(n+1),n)/(n+1).

Original entry on oeis.org

1, 1, 5, 55, 969, 23751, 749398, 28989675, 1329890705, 70625252863, 4263421511271, 288417894029200, 21616536107173175, 1778197364075525550, 159297460456229992380, 15438280311293473537331, 1609484153977526457766689, 179612918129148904884024975

Offset: 0

Paul D. Hanna, Dec 02 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..338
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A014068, A107863, A135860, A135862, A295763.

A135861:=n->binomial(n*(n+1),n)/(n+1); seq(A135861(n), n=0..15); # Wesley Ivan Hurt, May 08 2014

Table[Binomial[n*(n + 1), n]/(n + 1), {n, 0, 15}]

PARI
```
a(n)=binomial(n*(n+1),n)/(n+1)
```

a(n) = A135860(n)/(n+1).
a(n) = [x^(n^2)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017
a(p) == 1 ( mod p^4 ) for prime p >= 5 and a(2*p) == 4*p + 1 ( mod p^4 ) for prime p >= 5 (apply Mestrovic, equation 37). - Peter Bala, Feb 23 2020
a(n) ~ exp(n + 1/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Oct 17 2020

A135860 a(n) = binomial(n*(n+1), n).

Original entry on oeis.org

1, 2, 15, 220, 4845, 142506, 5245786, 231917400, 11969016345, 706252528630, 46897636623981, 3461014728350400, 281014969393251275, 24894763097057357700, 2389461906843449885700, 247012484980695576597296, 27361230617617949782033713, 3233032526324680287912449550

Offset: 0

Paul D. Hanna, Dec 02 2007

Seiichi Manyama, Table of n, a(n) for n = 0..337
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A014062, A014068, A054688, A107863, A135861, A135862.

[Binomial(n*(n+1), n): n in [0..30]]; // G. C. Greubel, Feb 20 2022

Table[Binomial[n^2 + n, n], {n, 0, 16}] (* Arkadiusz Wesolowski, Jul 18 2012 *)
(* or *)
Table[SeriesCoefficient[(1+x)^(n*(n+1)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

a(n)=binomial(n*(n+1),n)
for(n=0,15,print1(a(n),", "))

a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k))
for(n=0,15,print1(a(n),", "))

[binomial(n*(n+1), n) for n in (0..30)] # G. C. Greubel, Feb 20 2022

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k). - Paul D. Hanna, Nov 18 2015
a(n) is divisible by (n+1): a(n)/(n+1) = A135861(n).
a(n) is divisible by (n^2+1): a(n)/(n^2+1) = A135862(n).
a(n) = binomial(2*A000217(n),n). - Arkadiusz Wesolowski, Jul 18 2012
a(n) = [x^n] 1/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Oct 03 2017
a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Feb 08 2019
a(p) == p + 1 ( mod p^4 ) for prime p >= 5 and a(2*p) == (4*p + 1)*(2*p + 1) ( mod p^4 ) for all prime p. Apply Mestrovic, equation 37. - Peter Bala, Feb 27 2020
a(n) = ((n^2 + n)!)/((n^2)! * n!). - Peter Luschny, Feb 27 2020
a(n) = [x^n] (1 + x)^(n*(n+1)). - Vaclav Kotesovec, Aug 06 2025

cp's OEIS Frontend

A107862 Triangle, read by rows, where T(n,k) = C(n(n-1)/2 - k(k-1)/2 + n-k, n-k).

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A135861 a(n) = binomial(n*(n+1),n)/(n+1).

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Maple

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PARI

Formula

A135860 a(n) = binomial(n*(n+1), n).

Views

Author

Keywords

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Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

cp's OEIS Frontend

A107862 Triangle, read by rows, where T(n,k) = C(n*(n-1)/2 - k*(k-1)/2 + n-k, n-k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A135861 a(n) = binomial(n*(n+1),n)/(n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A135860 a(n) = binomial(n*(n+1), n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A107862 Triangle, read by rows, where T(n,k) = C(n(n-1)/2 - k(k-1)/2 + n-k, n-k).