A070967 -id:A070967 - cp's OEIS frontend

A070775 a(n) = Sum_{k=0..n} binomial(4n,4k).

1, 2, 72, 992, 16512, 261632, 4196352, 67100672, 1073774592, 17179738112, 274878431232, 4398044413952, 70368752566272, 1125899873288192, 18014398643699712, 288230375614840832, 4611686020574871552, 73786976286248271872, 1180591620751771041792, 18889465931341141901312

Offset: 0

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Also the cogrowth sequence of the 16-element group C4 X C4 = . - Sean A. Irvine, Nov 09 2024

Seiichi Manyama, Table of n, a(n) for n = 0..830
Index entries for linear recurrences with constant coefficients, signature (12,64).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), this sequence (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

a := n -> if n = 0 then 1 else 4^(n - 1)*(2*(-1)^n + 4^n) fi:
seq(a(n), n = 0..19); # Peter Luschny, Jul 02 2022

Table[Sum[Binomial[4n,4k],{k,0,n}],{n,0,30}] (* or *) Join[{1}, LinearRecurrence[{12,64},{2,72},30]] (* Harvey P. Dale, Apr 24 2011 *)

PARI
```
a(n)=sum(k=0,n,binomial(4*n,4*k))
```

N=66; x='x+O('x^N); Vec((1-10*x-16*x^2)/((1-16*x)*(1+4*x))) \\ Seiichi Manyama, Mar 15 2019

a(n) = (1/2)*(-4)^n + (1/4)*16^n for n > 0.
Let b(n) = a(n) - 2^(4n)/4 then b(n+1) = 4*b(n) - Benoit Cloitre, May 27 2004
G.f.: (1 - 10*x - 16*x^2)/((1-16*x)*(1+4*x)). - Seiichi Manyama, Mar 15 2019
G.f.: ((cos(x) + cosh(x))/2)^2 = Sum_{n >= 0} a(n)*x(4*n)/(4*n)!. - Peter Bala, Jun 20 2022

A070782 a(n) = Sum_{k=0..n} binomial(5n,5k).

Original entry on oeis.org

1, 2, 254, 6008, 215766, 6643782, 215492564, 6863694378, 219993856006, 7035859329512, 225191238869774, 7205634556190798, 230585685502492596, 7378682274243863442, 236118494435702913134, 7555784484021765207768, 241785184867484394069286, 7737125013254912900576822

Offset: 0

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Seiichi Manyama, Table of n, a(n) for n = 0..664
Index entries for linear recurrences with constant coefficients, signature (21,353,-32).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), this sequence (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

LinearRecurrence[{21,353,-32},{1,2,254},20] (* Harvey P. Dale, Jun 18 2023 *)

PARI
```
a(n)=sum(k=0,n,binomial(5*n,5*k))
```

Vec((1 - 19*x - 141*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)) + O(x^20)) \\ Colin Barker, May 27 2019

a(n) = (1/5)*32^n + (2/5)*(-11/2 + (5/2)*sqrt(5))^n + (2/5)*(-11/2 - (5/2)*sqrt(5))^n.
Let b(n) = a(n) - 2^(5n)/5; then b(n) + 11*b(n-1) - b(n-2) = 0. - Benoit Cloitre, May 27 2004
From Colin Barker, May 27 2019: (Start)
G.f.: (1 - 19*x - 141*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
(End)

A070832 a(n) = Sum_{k=0..n} binomial(8n,8k).

Original entry on oeis.org

1, 2, 12872, 1470944, 622116992, 125858012672, 36758056208384, 8793364151263232, 2334899414608412672, 586347560750962049024, 151652224498623981289472, 38612725801339748322639872, 9913426188311626771400228864

Offset: 0

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Seiichi Manyama, Table of n, a(n) for n = 0..415
Index entries for linear recurrences with constant coefficients, signature (136,32880,-552704,-65536).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), this sequence (b=8), A094213 (b=9), A070833 (b=10).

Table[Sum[Binomial[8n,8k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Nov 25 2020 *)

a(n)=sum(k=0,n,binomial(8*n,8*k)); \\ Benoit Cloitre, May 27 2004

Vec((1 - 134*x - 20280*x^2 + 207296*x^3 + 8192*x^4) / ((1 - 16*x)*(1 - 256*x)*(1 + 136*x + 16*x^2)) + O(x^15)) \\ Colin Barker, May 27 2019

Let b(n) = a(n)-2^(8*n)/8 then b(n)+120*b(n-1)-2160*b(n-2)-256*b(n-3)=0. - Benoit Cloitre, May 27 2004
a(n) = 1/4*16^n + 1/8*256^n + 1/4*(-68 + 48*sqrt(2))^n + 1/4*(-68-48*sqrt(2))^n.
From Colin Barker, May 27 2019: (Start)
G.f.: (1 - 134*x - 20280*x^2 + 207296*x^3 + 8192*x^4) / ((1 - 16*x)*(1 - 256*x)*(1 + 136*x + 16*x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>4.
(End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A070833 a(n) = Sum_{k=0..n} binomial(10n,10k).

Original entry on oeis.org

1, 2, 184758, 60090032, 139541849878, 94278969044262, 126648421364527548, 111019250117021378442, 125257104438594491956518, 121088185204450642433930072, 128442558588779813655233443038, 128767440665677943753184267342902

Offset: 0

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Seiichi Manyama, Table of n, a(n) for n = 0..332
Index entries for linear recurrences with constant coefficients, signature (522,587797,-75135226,-392963125,3200000).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), this sequence (b=10). Cf. A000032.

a(n) = sum(k=0, n, binomial(10*n, 10*k)); \\ Michel Marcus, Mar 15 2019

Vec((1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)) + O(x^15)) \\ Colin Barker, Mar 15 2019

a(n) = 1/10*1024^n+1/5*(-625/2+275/2*sqrt(5))^n+1/5*(-625/2-275/2*sqrt(5))^n+1/5*(123/2+55/2*sqrt(5))^n+1/5*(123/2-55/2*sqrt(5))^n.
G.f.: (1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)). - Colin Barker, Mar 15 2019
a(2*n) = (2^(20*n-1) + Lucas(20*n) + 5^(5*n)*Lucas(10*n))/5, for n>0 and for Lucas(n) = A000032(n). - Greg Dresden, Feb 04 2023

A094211 a(n) = Sum_{k=0..n} binomial(7n,7k).

Original entry on oeis.org

1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, 79450506979090, 10353832741654602, 1313930226050847938, 168883831255263816554, 21573903987107973878962, 2764126124873404346104778, 353643666623193292098680930, 45276535087893983968685884906

Offset: 0

Benoit Cloitre, May 27 2004

Seiichi Manyama, Table of n, a(n) for n = 0..474
Index entries for linear recurrences with constant coefficients, signature (71,7585,-36991,-128).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), this sequence (b=7), A070832 (b=8), A094213 (b=9), A070833 (b=10).

A094211:=n->add(binomial(7*n,7*k), k=0..n): seq(A094211(n), n=0..20); # Wesley Ivan Hurt, Feb 16 2017

Table[Sum[Binomial[7n,7k],{k,0,n}],{n,0,20}] (* or *) LinearRecurrence[ {71,7585,-36991,-128},{1,2,3434,232562},20] (* Harvey P. Dale, May 06 2012 *)

PARI
```
a(n)=sum(k=0,n,binomial(7*n,7*k))
```

Let b(n) = a(n) - 2^(7n)/7; then b(n) + 57*b(n-1) - 289*b(n-2) - b(n-3) = 0.
a(n) = 71*a(n-1) + 7585*a(n-2) - 36991*a(n-3) - 128*a(n-4); a(0)=1, a(1)=2, a(2)=3434, a(3)=232562. - Harvey P. Dale, May 06 2012
G.f.: (1 - 69*x - 4293*x^2 + 10569*x^3) / ((1 - 128*x)*(1 + 57*x - 289*x^2 - x^3)). - Colin Barker, May 27 2019
a(n) = (1 + 2*(s*(3 - 4*s^2))^(7*n) + 2*(-1)^n*((1 - 2*s^2)^(7*n) + s^(7*n))) * 2^(7*n)/7, where s = sin(Pi/14). - Vaclav Kotesovec, Apr 17 2023

A094213 a(n) = Sum_{k=0..n} binomial(9n,9k).

Original entry on oeis.org

1, 2, 48622, 9373652, 9263421862, 3433541316152, 2140802758677844, 984101481334553024, 536617781178725122150, 265166261617029717011822, 138567978655457801631498052, 70126939586658252408697345838, 36144812798331420987905742371116

Offset: 0

Benoit Cloitre, May 27 2004

Seiichi Manyama, Table of n, a(n) for n = 0..369
Index entries for linear recurrences with constant coefficients, signature (265,139823,-6826204,-6965249,512).

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), this sequence (b=9), A070833 (b=10).

Table[Sum[Binomial[9n,9k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Jul 14 2019 *)

PARI
```
a(n)=sum(k=0,n,binomial(9*n,9*k))
```

Vec((1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)) + O(x^15)) \\ Colin Barker, May 27 2019

Let b(n) = a(n)-2^(9*n)/9 then b(n)+246*b(n-1)-13605*b(n-2)+b(n-3)+(-1)^n*3078=0.
Conjectures from Colin Barker, May 27 2019: (Start)
G.f.: (1 - 263*x - 91731*x^2 + 3035380*x^3 + 1547833*x^4) / ((1 + x)*(1 - 512*x)*(1 + 246*x - 13605*x^2 + x^3)).
a(n) = 265*a(n-1) + 139823*a(n-2) - 6826204*a(n-3) - 6965249*a(n-4) + 512*a(n-5) for n>4. (End)
a(n) ~ (1/9)*exp(n*9*log(2)) (conjecture). - Bill McEachen, Aug 11 2025

A361043 Array read by descending antidiagonals. A(n, k) is, if n > 0, the number of multiset permutations of {0, 1} of length n * k where the number of occurrences of 1 are multiples of n. A(0, k) = k + 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 8, 8, 2, 1, 6, 16, 32, 22, 2, 1, 7, 32, 128, 170, 72, 2, 1, 8, 64, 512, 1366, 992, 254, 2, 1, 9, 128, 2048, 10922, 16512, 6008, 926, 2, 1, 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1, 11, 512, 32768, 699050, 4196352, 6643782, 2973350, 232562, 12872, 2, 1

Offset: 0

Peter Luschny, Mar 18 2023

Because of the interchangeability of 0 and 1 in the definition, A(n, k) is even if n, k >= 1. In other words, if the binary representation of a permutation of the defined type is counted, then so is the 1's complement of that representation.

			Array A(n, k) starts:
 [0] 1, 2,    3,      4,        5,          6,            7, ...  A000027
 [1] 1, 2,    4,      8,       16,         32,           64, ...  A000079
 [2] 1, 2,    8,     32,      128,        512,         2048, ...  A081294
 [3] 1, 2,   22,    170,     1366,      10922,        87382, ...  A007613
 [4] 1, 2,   72,    992,    16512,     261632,      4196352, ...  A070775
 [5] 1, 2,  254,   6008,   215766,    6643782,    215492564, ...  A070782
 [6] 1, 2,  926,  37130,  2973350,  174174002,  11582386286, ...  A070967
 [7] 1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, ...  A094211
.
Triangle T(n, k) starts:
 [0]  1;
 [1]  2,   1;
 [2]  3,   2,    1;
 [3]  4,   4,    2,     1;
 [4]  5,   8,    8,     2,      1;
 [5]  6,  16,   32,    22,      2,      1;
 [6]  7,  32,  128,   170,     72,      2,     1;
 [7]  8,  64,  512,  1366,    992,    254,     2,    1;
 [8]  9, 128, 2048, 10922,  16512,   6008,   926,    2, 1;
 [9] 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1;
.
A(2, 2) = 8 = card(0000, 1100, 1010, 1001, 0110, 0101, 0011, 1111).
A(1, 3) = 8 = card(000, 100, 010, 001, 110, 101, 011, 111).

Rows: A000027 (n=0), A000079 (n=1), A081294 (n=2), A007613 (n=3), A070775 (n=4), A070782 (n=5), A070967 (n=6), A094211 (n=7), A070832 (n=8), A094213 (n=9), A070833 (n=10).
Variant: A308500 (upwards antidiagonals).
Cf. A167009 (main diagonal).

T := (n, k) -> add(binomial((n - k)*k, j*k), j = 0 .. n-k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);

# In Python use this import:
# from sympy.utilities.iterables import multiset_permutations
def A(n: int, k: int) -> int:
    if n == 0: return k + 1
    count = 0
    for a in range(0, n * k + 1, n):
        S = [i < a for i in range(n * k)]
        count += Permutations(S).cardinality()
    return count
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size)]
for n in range(6): print(ARow(n, 5))

A(n, k) = Sum_{j=0..k} binomial(n*k, n*j).

T(n, k) = Sum_{j=0..n-k} binomial((n - k)*k, j*k).

A308500 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=0..n} binomial(kn,kj).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 8, 8, 5, 1, 2, 22, 32, 16, 6, 1, 2, 72, 170, 128, 32, 7, 1, 2, 254, 992, 1366, 512, 64, 8, 1, 2, 926, 6008, 16512, 10922, 2048, 128, 9, 1, 2, 3434, 37130, 215766, 261632, 87382, 8192, 256, 10, 1, 2, 12872, 232562, 2973350, 6643782, 4196352, 699050, 32768, 512, 11

Offset: 0

Seiichi Manyama, Jun 01 2019

			Square array begins:
   1,  1,    1,     1,       1,         1, ...
   2,  2,    2,     2,       2,         2, ...
   3,  4,    8,    22,      72,       254, ...
   4,  8,   32,   170,     992,      6008, ...
   5, 16,  128,  1366,   16512,    215766, ...
   6, 32,  512, 10922,  261632,   6643782, ...
   7, 64, 2048, 87382, 4196352, 215492564, ...

Seiichi Manyama, Antidiagonals n = 0..115, flattened

Columns k=0..10 give A000027(n+1), A000079, A081294, A007613, A070775, A070782, A070967, A094211, A070832, A094213, A070833.

Main diagonal gives A167009.

T[n_, k_] := Sum[Binomial[k*n, k*j], {j, 0, n}] ; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A177810 Triangle binomial(6n,6m), 0 <= m <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 924, 1, 1, 18564, 18564, 1, 1, 134596, 2704156, 134596, 1, 1, 593775, 86493225, 86493225, 593775, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 5245786, 11058116888, 353697121050, 353697121050, 11058116888, 5245786, 1

Offset: 0

Roger L. Bagula, Dec 13 2010

Row sums are A070967. k=6 in binomial(k*n,k*m) sequence similar to k=2 in A086645, k=4 in A070775,...

			1;
1, 1;
1, 924, 1;
1, 18564, 18564, 1;
1, 134596, 2704156, 134596, 1;
1, 593775, 86493225, 86493225, 593775, 1;

Cf. A034839, A177808, A139459, A070782, A177809, A070967.

t[n_, m_] := Binomial[n, 6*m]; Flatten@Table[Table[t[n, m], {m, 0, n/6}], {n, 0, 42, 6}]

Left-right symmetric: binomial(6*n,6*m) = binomial(6*n,6*(n-m)).

cp's OEIS Frontend

A070775 a(n) = Sum_{k=0..n} binomial(4*n,4*k).

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A070782 a(n) = Sum_{k=0..n} binomial(5*n,5*k).

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A070832 a(n) = Sum_{k=0..n} binomial(8*n,8*k).

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A070833 a(n) = Sum_{k=0..n} binomial(10*n,10*k).

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A094211 a(n) = Sum_{k=0..n} binomial(7*n,7*k).

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A094213 a(n) = Sum_{k=0..n} binomial(9*n,9*k).

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A361043 Array read by descending antidiagonals. A(n, k) is, if n > 0, the number of multiset permutations of {0, 1} of length n * k where the number of occurrences of 1 are multiples of n. A(0, k) = k + 1.

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Formula

A070775 a(n) = Sum_{k=0..n} binomial(4n,4k).

A070782 a(n) = Sum_{k=0..n} binomial(5n,5k).

A070832 a(n) = Sum_{k=0..n} binomial(8n,8k).

A070833 a(n) = Sum_{k=0..n} binomial(10n,10k).

A094211 a(n) = Sum_{k=0..n} binomial(7n,7k).

A094213 a(n) = Sum_{k=0..n} binomial(9n,9k).

A308500 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=0..n} binomial(kn,kj).

A177810 Triangle binomial(6n,6m), 0 <= m <= n, read by rows.