A070775 -id:A070775 - cp's OEIS frontend

A216357 Expansion of 1/( (1-16x)(1+4*x)^2 )^(1/4).

1, 2, 38, 404, 5510, 74492, 1048924, 15004776, 217943238, 3200089580, 47405806708, 707305846936, 10616181542044, 160142807848792, 2426097698458360, 36890818642990544, 562772826273060678, 8609639617006367052, 132048790603779592196, 2029851945081220214200

Offset: 0

Paul D. Hanna, Sep 04 2012

nonn

			G.f.: A(x) = 1 + 2*x + 38*x^2 + 404*x^3 + 5510*x^4 + 74492*x^5 + 1048924*x^6 + ...
where 1/A(x)^4 = 1 - 8*x - 112*x^2 - 256*x^3.
The logarithm of the g.f. begins:
log(A(x)) = x + 2*x^2/2 + 72*x^3/3 + 992*x^4/4 + 16512*x^5/5 + 261632*x^6/6 + 4196352*x^7/7 + ... + A070775(n)*x^n/n + ...

Robert Israel, Table of n, a(n) for n = 0..830

Cf. A070775, A216316, A216358.

f:= gfun:-rectoproc({(48+64*n)*a(n)+(14+12*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 2}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Dec 09 2016

a = DifferenceRoot[Function[{a, n}, {(48+64n) a[n] + (14+12n) a[1+n] + (-2-n) a[2+n] == 0, a[0] == 1, a[1] == 2}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 16 2022, after Robert Israel *)

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(4*m, 4*j))*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

G.f.: exp(Sum_{n>=1} A070775(n)*x^n/n) where A070775(n) = Sum_{k=0..n} binomial(4*n,4*k).
a(n) ~ GAMMA(3/4) * 2^(4*n+1/2) / (Pi* sqrt(5) * n^(3/4)). - Vaclav Kotesovec, Jul 31 2014
a(n) = ((64*n-80)*a(n-2)+(12*n-10)*a(n-1))/n. - Robert Israel, Dec 09 2016

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).

A377855 Cogrowth sequence of the 16-element group C4:C4 = .

Original entry on oeis.org

1, 0, 2, 6, 40, 120, 512, 2016, 8320, 32640, 131072, 523776, 2099200, 8386560, 33554432, 134209536, 536903680, 2147450880, 8589934592, 34359607296, 137439477760, 549755289600, 2199023255552, 8796090925056, 35184380477440, 140737479966720, 562949953421312

Offset: 0

Sean A. Irvine, Nov 09 2024

Gives the even terms, all the odd terms are 0.

Index entries for linear recurrences with constant coefficients, signature (2, 4, 8, 32).

Cf. A070775 (C4 X C4), A377840 (C8 X C2).

G.f.: (12*x^4+6*x^3+2*x^2+2*x-1) / ((4*x-1) * (2*x+1) * (4*x^2+1)).

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 *)

A177809 Symmetrical sequence:Binomial(n,5*m).

Original entry on oeis.org

1, 1, 1, 1, 252, 1, 1, 3003, 3003, 1, 1, 15504, 184756, 15504, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 142506, 30045015, 155117520, 30045015, 142506, 1, 1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1, 1, 658008, 847660528, 40225345056

Offset: 0

Roger L. Bagula, Dec 13 2010

Row sums are A070782.

5th in the sequence of sequence Binomial(n,k*m),k=1,2,3,4,5,...

			{1},
{1, 1},
{1, 252, 1},
{1, 3003, 3003, 1},
{1, 15504, 184756, 15504, 1},
{1, 53130, 3268760, 3268760, 53130, 1},
{1, 142506, 30045015, 155117520, 30045015, 142506, 1},
{1, 324632, 183579396, 3247943160, 3247943160, 183579396, 324632, 1},
{1, 658008, 847660528, 40225345056, 137846528820, 40225345056, 847660528, 658008, 1},
{1, 1221759, 3190187286, 344867425584, 3169870830126, 3169870830126, 344867425584, 3190187286, 1221759, 1},
{1, 2118760, 10272278170, 2250829575120, 47129212243960, 126410606437752, 47129212243960, 2250829575120, 10272278170, 2118760, 1}

Cf. A086645, A034839, A070775, A177808, A139459, A070782.

t[n_, m_] = Binomial[n, 5*m];
Table[Table[t[n, m], {m, 0, Floor[n/5]}], {n, 0, 50, 5}];
Flatten[%]

A377843 Cogrowth sequence of the 16-element group C4 X C2 X C2 = .

Original entry on oeis.org

1, 2, 9, 62, 689, 7322, 69369, 616982, 5422049, 48197042, 433434729, 3913915502, 35311723409, 317999340362, 2860994944089, 25738114039622, 231602961592769, 2084457277181282, 18761300850805449, 168858054223133342, 1519730933499158129, 13677470410291063802

Offset: 0

Sean A. Irvine, Nov 09 2024

Gives the even terms, all the odd terms are 0.

			a(2)=9 corresponds to the words SSSS, TTTT, UUUU, TTUU, TUUT, UUTT, TUTU, UTUT, UTTU.

Index entries for linear recurrences with constant coefficients, signature (15,-78,210,79,-225).

Cf. A070775 (C4 X C4), A377714 (C4 X C2), A377840 (C8 X C2), A007582 (D8).

LinearRecurrence[{15,-78,210,79,-225},{1, 2, 9, 62, 689},22] (* James C. McMahon, Nov 10 2024 *)

G.f.: (38*x^4+127*x^3-57*x^2+13*x-1) / ((1-x) * (9*x-1) * (x+1) * (25*x^2-6*x+1)).

E.g.f.: (2*exp(3*x)*cos(4*x) + 5*cosh(x) + cosh(9*x) + sinh(x) + sinh(9*x))/8. - Stefano Spezia, Nov 10 2024

A377943 Cogrowth sequence of the 16-element Pauli group C4 o D4 = .

Original entry on oeis.org

1, 1, 11, 91, 821, 7381, 66431, 597871, 5380841, 48427561, 435848051, 3922632451, 35303692061, 317733228541, 2859599056871, 25736391511831, 231627523606481, 2084647712458321, 18761829412124891, 168856464709124011, 1519708182382116101, 13677373641439044901

Offset: 0

Sean A. Irvine, Nov 11 2024

Gives the even terms, all the odd terms are 0.

Index entries for linear recurrences with constant coefficients, signature (9, 1, -9).

Cf. A070775 (C4 X C4), A377855 (C4:C4), A047854 (D4 X C2).

G.f.: (x^2-8*x+1) / ((x-1) * (9*x-1) * (x+1)).

A090411 Expansion of g.f. (1-x)/(1-16*x).

Original entry on oeis.org

1, 15, 240, 3840, 61440, 983040, 15728640, 251658240, 4026531840, 64424509440, 1030792151040, 16492674416640, 263882790666240, 4222124650659840, 67553994410557440, 1080863910568919040, 17293822569102704640, 276701161105643274240, 4427218577690292387840

Offset: 0

Paul Barry, Nov 30 2003

Index entries for linear recurrences with constant coefficients, signature (16).

Cf. A001025, A070775, A090407, A090408, A090409.

Join[{1, a = 15}, Table[a=16*a, {n,0,30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Join[{1},NestList[16#&,15,20]] (* Harvey P. Dale, Dec 28 2016 *)

a(n)=if(n,15<<(4*n-4),1) \\ Charles R Greathouse IV, Jun 10 2011

a(n) = 15*16^(n-1) + 0^n/16.
a(n) = Sum_{j=0..3, Sum_{k=0..n, C(4*n+j, 4*k)}}.
a(n) = (A070775(n) + A090407(n) + A001025(n) + A090408(n))/4.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (15*exp(16*x) + 1)/16.
a(n) = 16*a(n-1) for n > 1. (End)

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

A216357 Expansion of 1/( (1-16*x)*(1+4*x)^2 )^(1/4).

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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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A377855 Cogrowth sequence of the 16-element group C4:C4 = .

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

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A177809 Symmetrical sequence:Binomial(n,5*m).

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A377843 Cogrowth sequence of the 16-element group C4 X C2 X C2 = .

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A377943 Cogrowth sequence of the 16-element Pauli group C4 o D4 = .

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A090411 Expansion of g.f. (1-x)/(1-16*x).

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

A216357 Expansion of 1/( (1-16x)(1+4*x)^2 )^(1/4).

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.