A010763 -id:A010763 - cp's OEIS frontend

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

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

Clark Kimberling

			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

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.

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

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

A350653 a(n) is the number of weak compositions of n into n-1 parts in which at least one part is zero.

Original entry on oeis.org

0, 2, 12, 52, 205, 786, 2996, 11432, 43749, 167950, 646635, 2496132, 9657687, 37442146, 145422660, 565722704, 2203961413, 8597496582, 33578000591, 131282408380, 513791607399, 2012616400058, 7890371113927, 30957699535752, 121548660036275

Offset: 2

Enrique Navarrete, Jan 09 2022

			a(5)=52 since 5 can be written as 5+0+0+0 (4 such compositions); 4+1+0+0 (12 such compositions); 3+2+0+0 (12 such compositions); 3+1+1+0 (12 such compositions); 2+2+1+0 (12 such compositions). All these weak compositions contain at least one zero.

Cf. A001791, A010763.

a[n_] := Binomial[2*n - 2, n] - n + 1; Array[a, 25, 2] (* Amiram Eldar, Jan 10 2022 *)

a(n) = binomial(2*n-2,n) - (n-1) = A001791(n-1) -n+1.
G.f.: 4*x^2/((1 - sqrt(1 - 4*x))^2*sqrt(1 - 4*x)) - (1 - 2*x + 2*x^2)/(1 - x)^2. - Stefano Spezia, Jan 10 2022
D-finite with recurrence +n*(11*n-38)*a(n) -(n-1)*(73*n-244)*a(n-1) +2*(67*n^2-364*n+492)*a(n-2) -4*(9*n-22)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Mar 06 2022

A375178 a(n) = Sum_{k = 0..n-1} binomial(n+k-1, k)^3 (same as A112028 with an extra 0 at the start).

Original entry on oeis.org

0, 1, 9, 244, 9065, 389376, 18188478, 897376152, 46011772521, 2427553965160, 130930630643384, 7186614533569296, 400132290102421214, 22543708920891189136, 1282873288801683197250, 73628947696550668509744, 4257138240245923453355625, 247733479854085081062353400, 14498252738780732999484606360

Offset: 0

Peter Bala, Aug 03 2024

Compare with the identity Sum_{k = 0..n-1} binomial(n+k-1, k) = (1/2) * binomial(2*n, n) = (1/2) * A000984(n) for n >= 1.
The central binomial coefficients satisfy the supercongruence (1/2) * binomial(2*p, p) == 1 (mod p^3) for all primes p >= 5 (Wolstenholme's theorem).
For prime p, binomial(p+k-1, k) == 0 (mod p) for 1 <= k <= p-1. It follows that a(p) == 1 (mod p^3) for all primes p. We conjecture that, in fact, the stronger congruence a(p) == 1 (mod p^5) holds for all primes p >= 7.
Further, we conjecture that for r >= 2 and prime p >= 5, a(p^r) == a(p^(r-1)) (mod p^(3*r+3)).
More generally, for a positive integer m, define a sequence {b_m(n) : n >= 0} by setting b_m(n) = Sum_{k = 0..n-1} binomial(n+k-1, k)^(2*m+1). Then the congruence b_m(p) == 1 (mod p^(2*m+1)) clearly holds for all primes p. We conjecture that the stronger supercongruence b_m(p) == 1 (mod p^(2*m+3)) holds for all primes p >= 2*m + 5, and for r >= 2, the supercongruence b_m(p^r) == b_m(p^(r-1)) (mod p^(3*r+2*m+1)) also holds for all primes p >= 2*m + 5.
Essentially a duplicate of A112028.

			Examples of supercongruences:
a(7) - a(1) = 897376152 - 1 = (7^5)*107*499 == 0 (mod 7^5)
a(11) - a(1) = 7186614533569296 - 1 = 5*(11^5)*8924644409 == 0 (mod 11^5).

Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).

Cf. A000984, A010763, A112028, A176335, A375179, A375180.

seq(add( binomial(n+k-1, k)^3, k = 0..n-1), n = 0..20);

a(n) = Sum_{k = 0..n-1} (-1)^k * binomial(-n, k)^3.

a(n) ~ 2^(6*n-3)/(7*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Aug 03 2024

A371003 a(n) = binomial(2n-1,n) - binomial(n,2)(binomial(n-1,2) + 2) - 1.

Original entry on oeis.org

0, 0, 0, 4, 45, 281, 1358, 5790, 23229, 90667, 350130, 1348315, 5194995, 20051019, 77548994, 300527354, 1166786517, 4537546535, 17672605394, 68923231539, 269128896899, 1052049432887, 4116715304850, 16123801771169, 63205303135475, 247959266375901, 973469712709278

Offset: 1

Enrique Navarrete, Mar 07 2024

nonn

a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 3 boxes remaining empty.

a(n) is also the number of weak compositions of n into n parts in which at least three parts are zero.

			a(5)=45 since 5 can be written as 5+0+0+0+0, 0+5+0+0+0, etc. (5 such compositions); 4+1+0+0+0 (20 such compositions); 3+2+0+0+0 (20 such compositions).

Cf. A010763, A352027, A359435.

Table[Binomial[2n-1,n]-Binomial[n,2]*(Binomial[n-1,2]+2)-1,{n,27}] (* James C. McMahon, Mar 08 2024 *)

from math import comb
def A371003(n): return comb((n<<1)-1,n)-n-((m:=(n-1)**2)*(m+3)>>2) # Chai Wah Wu, Mar 29 2024

A375555 Triangle read by rows: T(n, k) = abs(A181937(k, n)), where A181937 are the André numbers, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 16, 9, 4, 1, 1, 1, 61, 19, 14, 5, 1, 1, 1, 272, 99, 34, 20, 6, 1, 1, 1, 1385, 477, 69, 55, 27, 7, 1, 1, 1, 7936, 1513, 496, 125, 83, 35, 8, 1, 1, 1, 50521, 11259, 2896, 251, 209, 119, 44, 9, 1

Offset: 0

Peter Luschny, Aug 19 2024

See A181937 for comments and references.

			Triangle starts:
  [0]  1;
  [1]  1, 1;
  [2]  1, 1,    1;
  [3]  1, 1,    2,    1;
  [4]  1, 1,    5,    3,   1;
  [5]  1, 1,   16,    9,   4,   1;
  [6]  1, 1,   61,   19,  14,   5,  1;
  [7]  1, 1,  272,   99,  34,  20,  6,  1;
  [8]  1, 1, 1385,  477,  69,  55, 27,  7, 1;
  [9]  1, 1, 7936, 1513, 496, 125, 83, 35, 8, 1;
.
Seen as an array:
  [0]  1, 1,      1,      1,      1,      1,      1,      1, ...
  [1]  1, 1,      2,      3,      4,      5,      6,      7, ...
  [2]  1, 1,      5,      9,     14,     20,     27,     35, ...
  [3]  1, 1,     16,     19,     34,     55,     83,    119, ...
  [4]  1, 1,     61,     99,     69,    125,    209,    329, ...
  [5]  1, 1,    272,    477,    496,    251,    461,    791, ...
  [6]  1, 1,   1385,   1513,   2896,   2300,    923,   1715, ...
  [7]  1, 1,   7936,  11259,  11056,  15775,  10284,   3431, ...

Cf. A181937, A375554 (row sums), A030662 (central terms, main diagonal of array), A010763 (central terms of the (1, 1)-based variant).

Andre := proc(n, k) option remember; local j;
  ifelse(k = 0, 1, ifelse(n = 0, 1,
  -add(binomial(k, j) * Andre(n, j), j = 0..k-1, n))) end:
T := (n, k) -> abs(Andre(k, n)): seq(seq(T(n, k), k = 0..n), n = 0..10);

Andre[n_, k_] := Andre[n, k] = If[k <= 0, 1, If[n == 0, 1, -Sum[Binomial[k, j] Andre[n, j], {j, 0, k-1, n}]]];
(* Seen as an array: *)
A[n_, k_] := Abs[Andre[k, n + k]];
Table[A[n, k], {n, 0, 9}, {k, 0, 7}] // MatrixForm

A352405 a(n) = binomial(n,2)*(binomial(n-1,2) + 2).

Original entry on oeis.org

0, 2, 9, 30, 80, 180, 357, 644, 1080, 1710, 2585, 3762, 5304, 7280, 9765, 12840, 16592, 21114, 26505, 32870, 40320, 48972, 58949, 70380, 83400, 98150, 114777, 133434, 154280, 177480, 203205, 231632, 262944, 297330, 334985, 376110, 420912, 469604, 522405, 579540, 641240, 707742, 779289, 856130

Offset: 1

Enrique Navarrete, Mar 14 2022

a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with either 1 or 2 boxes remaining empty.
a(n) is also the number of weak compositions of n into n parts that contain either one or two 0's.
a(n)+1 is the number of ways to place n indistinguishable balls into n distinguishable boxes with at most 2 boxes remaining empty (just add the case of no empty boxes in which we place exactly one ball in one box).

			a(4)=30 since 4 can be written as 3+1+0+0, 0+3+0+1, etc. (12 such compositions); 2+2+0+0 (6 such compositions); 2+1+1+0 (12 such compositions).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A010763, A350653.

a[n_] := Binomial[n, 2] * (Binomial[n - 1, 2] + 2); Array[a, 50] (* Amiram Eldar, Mar 15 2022 *)

G.f.: x^2*(2 - x + 5*x^2)/(1 - x)^5. - Stefano Spezia, Mar 15 2022

A370197 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 4 boxes remaining empty.

Original entry on oeis.org

0, 0, 0, 0, 5, 81, 658, 3830, 18525, 80587, 330330, 1312015, 5132075, 19946915, 77383374, 300272554, 1166405717, 4536991655, 17671814690, 68922126879, 269127380699, 1052047384687, 4116712577510, 16123798186665, 63205298480275, 247959260395901, 973469705104278

Offset: 1

Enrique Navarrete, Mar 09 2024

nonn

a(n) is also the number of weak compositions of n into n parts in which at least four parts are zero.

			a(6)=81 since 6 can be written as 6+0+0+0+0+0, 0+6+0+0+0+0, etc. (6 such compositions); 5+1+0+0+0+0 (30 such compositions); 4+2+0+0+0+0 (30 such compositions); 3+3+0+0+0+0 (15 such compositions).

Cf. A001700, A010763, A352027, A371003.

a(n) = binomial(2*n-1,n) - binomial(n,2)*binomial(n-1,2) - binomial(n,3)*binomial(n-1,3) - n*(n-1) - 1.

A371036 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least one box remaining empty and not all balls placed in a single box.

Original entry on oeis.org

0, 0, 6, 30, 120, 455, 1708, 6426, 24300, 92367, 352704, 1352065, 5200286, 20058285, 77558744, 300540178, 1166803092, 4537567631, 17672631880, 68923264389, 269128937198, 1052049481837, 4116715363776, 16123801841525, 63205303218850, 247959266474025, 973469712824028

Offset: 1

Enrique Navarrete, Mar 08 2024

nonn

a(n) is also the number of weak compositions of n into n parts in which at least one part is zero and the composition does not contain a single nonzero part.

			a(4)=30 since 4 can be written as 3+1+0+0, 0+3+1+0, etc. (12 such compositions); 2+2+0+0 (6 such compositions); 2+1+1+0 (12 such compositions).

Cf. A001700, A010763, A048775, A352027, A359435.

a(n) = binomial(2n-1,n)-n-1, n > 1; a(1)=0.
a(n) = A048775(n-1)-1, n > 1.
a(n) = A001700(n-1)-(n+1), n > 1.

cp's OEIS Frontend

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

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A350653 a(n) is the number of weak compositions of n into n-1 parts in which at least one part is zero.

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A375178 a(n) = Sum_{k = 0..n-1} binomial(n+k-1, k)^3 (same as A112028 with an extra 0 at the start).

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A371003 a(n) = binomial(2*n-1,n) - binomial(n,2)*(binomial(n-1,2) + 2) - 1.

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A375555 Triangle read by rows: T(n, k) = abs(A181937(k, n)), where A181937 are the André numbers, for 0 <= k <= n.

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A352405 a(n) = binomial(n,2)*(binomial(n-1,2) + 2).

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A370197 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 4 boxes remaining empty.

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A371036 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least one box remaining empty and not all balls placed in a single box.

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Author

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Comments

Examples

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Formula

A371003 a(n) = binomial(2n-1,n) - binomial(n,2)(binomial(n-1,2) + 2) - 1.