A120305 -id:A120305 - cp's OEIS frontend

A307318 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).

1, -2, 37, -692, 14371, -315002, 7156969, -166785320, 3960790687, -95442311582, 2326713829837, -57260397539204, 1420295354815351, -35463581316556850, 890530353765972817, -22472131364683145552, 569507678494598796631, -14487492070374441746150

Offset: 0

Seiichi Manyama, Apr 02 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A120305, A144660, A307324.

Table[Sum[(-1)^(i + j + k) * (i + j + k)!/(i!*j!*k!), {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)

{a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, (-1)^(i+j+k)*(i+j+k)!/(i!*j!*k!))))}

{a(n) = sum(i=0, 3*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^3, i))} \\ Seiichi Manyama, May 20 2019

From Vaclav Kotesovec, Apr 02 2019: (Start)
Recurrence: 6*(n-1)*n^2*(490*n^4 - 3948*n^3 + 11668*n^2 - 14967*n + 7027)*a(n) =  - (n-1)*(74480*n^6 - 675066*n^5 + 2399756*n^4 - 4233492*n^3 + 3852029*n^2 - 1682577*n + 272160)*a(n-1) + (131320*n^7 - 1437814*n^6 + 6472114*n^5 - 15414556*n^4 + 20770423*n^3 - 15610855*n^2 + 5939868*n - 861840)*a(n-2) - (27440*n^7 - 355838*n^6 + 1853810*n^5 - 4998800*n^4 + 7460459*n^3 - 6071312*n^2 + 2439561*n - 362880)*a(n-3) - 3*(2*n - 5)*(3*n - 8)*(3*n - 7)*(490*n^4 - 1988*n^3 + 2764*n^2 - 1515*n + 270)*a(n-4).
a(n) ~ (-1)^n * 3^(3*n + 7/2) / (128*Pi*n). (End)

A307324 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (-1)^(i+j+k+l) * (i+j+k+l)!/(i!j!k!*l!).

Original entry on oeis.org

1, 9, 997, 148041, 25413205, 4744544613, 935728207597, 191813392024137, 40462946725744501, 8726529512888314245, 1915408781755211655133, 426478330303800465141669, 96092667172064808771832957, 21869171662479233922632691261

Offset: 0

Seiichi Manyama, Apr 02 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Vaclav Kotesovec, Recurrence (of order 5)

Cf. A120305, A144661, A307318.

Table[Sum[(-1)^(i + j + k + l) * (i + j + k + l)! / (i!*j!*k!*l!), {i, 0, n}, {j, 0, n}, {k, 0, n}, {l, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)
Table[Sum[((-1)^(j + k + l) * 2^(-1 - j - k - l) * ((j + k + l)! * (1 + n)! + (-1)^n * 2^(1 + j + k + l) * (1 + j + k + l + n)! Hypergeometric2F1[1, 2 + j + k + l + n, 2 + n, -1]))/(j! k! l! (1 + n)!), {j, 0, n}, {k, 0, n}, {l, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2019 *)

{a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, sum(l=0, n, (-1)^(i+j+k+l)*(i+j+k+l)!/(i!*j!*k!*l!)))))}

{a(n) = sum(i=0, 4*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^4, i))} \\ Seiichi Manyama, May 20 2019

a(n) ~ 2^(8*n + 15/2) / (625 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 03 2019

A307349 a(n) = Sum_{i=1..n} Sum_{j=1..n} (-1)^(i+j) * (i+j)!/(2!i!j!).

Original entry on oeis.org

0, 1, 1, 5, 15, 56, 203, 757, 2839, 10736, 40821, 155948, 598065, 2301118, 8878591, 34340085, 133100055, 516851528, 2010358061, 7831136920, 30546063745, 119291436738, 466379022561, 1825168170620, 7149316835465, 28027993191706, 109965636641173

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1665

Cf. A048775, A120305, A307350, A307351.

Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(2*i!*j!), {i, 1, n}], {j, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 03 2019 *)

{a(n) = sum(i=1, n, sum(j=1, n, (-1)^(i+j)*(i+j)!/(2*i!*j!)))}

{a(n) = sum(i=2, 2*n, (-1)^i*i!*polcoef(sum(j=1, n, x^j/j!)^2, i))/2} \\ Seiichi Manyama, May 20 2019

a(n) = (A120305(n) - (-1)^n)/2. - Vaclav Kotesovec, Apr 03 2019
a(n) ~ 2^(2*n+1) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 03 2019
G.f.: (1/sqrt(1-4*z)-1+2*z/(1-z^2))/(2*(2+z)). - Sergey Perepechko, Jul 11 2019

A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

Original entry on oeis.org

1, 2, 6, 19, 65, 231, 841, 3110, 11628, 43834, 166298, 634140, 2428336, 9331688, 35967462, 138987715, 538287881, 2088842463, 8119916647, 31613327405, 123251518641, 481125828853, 1880262896537, 7355767408395, 28803717914791, 112887697489907, 442784607413427

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1664

Partial sums of A026641. - Seiichi Manyama, Jan 30 2023
Cf. A000108, A006134, A079309, A105872, A120305, A306409.
Cf. A191993, A360186, A360212.

Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(i!*j!), {i, 0, j}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 04 2019 *)

a(n) = sum(i=0, n, sum(j=i, n, (-1)^(i+j)*(i+j)!/(i!*j!)));

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 29 2023

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^3*(2/(1+sqrt(1-4*x)))^3))) \\ Seiichi Manyama, Jan 29 2023

a(n) = (A006134(n) + A120305(n))/2.
From Vaclav Kotesovec, Apr 04 2019: (Start)
Recurrence: 2*n*a(n) = (9*n-4)*a(n-1) - (3*n-2)*a(n-2) - 2*(2*n-1)*a(n-3).
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). (End)
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)^3) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

A371798 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-2k-1,n-2*k).

Original entry on oeis.org

1, 1, 2, 7, 26, 96, 356, 1331, 5014, 19006, 72412, 277058, 1063856, 4097510, 15823432, 61245987, 237536326, 922906150, 3591500972, 13996328322, 54614894396, 213360770840, 834409399672, 3266370155262, 12797894251276, 50184309630196, 196936674150296

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Sergi Elizalde, Nadia Lafrenière, Joel Brewster Lewis, Erin McNicholas, Jessica Striker, and Amanda Welch, Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks, arXiv:2412.16368 [math.CO], 2024. See p. 13.

Cf. A026641, A120305.

Table[Sum[(-1)^k Binomial[2n-2k-1,n-2k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Oct 31 2024 *)

a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k-1, n-2*k));

a(n) = [x^n] 1/((1+x^2) * (1-x)^n).
a(n) = binomial(2*n-1, n)*hypergeom([1, (1-n)/2, -n/2], [1/2-n, 1-n], -1). - Stefano Spezia, Apr 06 2024
a(n) ~ 2^(2*n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 07 2024
Conjectured g.f.: 1 + x*(4 - 10*x + 8*x^2)/(2 - 11*x + 14*x^2 - 8*x^3 + (2 - 3*x)*sqrt(1 - 4*x)) (see Elizalde et al. at p. 13). - Stefano Spezia, Dec 27 2024

A306409 a(n) = -Sum_{0<=i

Original entry on oeis.org

0, 1, 3, 10, 34, 120, 434, 1597, 5949, 22363, 84655, 322245, 1232205, 4729453, 18210279, 70307546, 272087770, 1055139408, 4099200524, 15951053566, 62159391150, 242542955378, 947504851414, 3705431067156, 14505084243860, 56831711106496, 222853334131080

Offset: 0

Seiichi Manyama, Apr 05 2019

nonn

			n | a(n) | A307354 | A006134 | A120305
--+------+---------+---------+---------
0 |    0 |       1 |       1 |       1
1 |    1 |       2 |       3 |       1
2 |    3 |       6 |       9 |       3
3 |   10 |      19 |      29 |       9
4 |   34 |      65 |      99 |      31
5 |  120 |     231 |     351 |     111

Seiichi Manyama, Table of n, a(n) for n = 0..1664

Partial sums of A014300. - Seiichi Manyama, Jan 30 2023

Cf. A000108, A000957, A006134, A057552, A120305, A307354.

Table[-Sum[Sum[(-1)^(i+j) * (i+j)!/(i!*j!), {i, 0, j-1}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 05 2019 *)

a(n) = -sum(i=0, n, sum(j=i+1, n, (-1)^(i+j)*(i+j)!/(i!*j!)));

my(N=30, x='x+O('x^N)); concat(0, Vec((1-sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x))))) \\ Seiichi Manyama, Jan 30 2023

a(n) = A006134(n) - A307354(n).
a(n) = (A006134(n) - A120305(n))/2.
a(n) ~ 4^(n+1) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 05 2019
G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( x *c(x)/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023

A308322 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} (-1)^(i_1 + i_2 + ... + i_k) * multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -2, 3, 0, 1, 1, 9, 37, 9, 1, 1, 1, -44, 997, -692, 31, 0, 1, 1, 265, 44121, 148041, 14371, 111, 1, 1, 1, -1854, 2882071, -66211704, 25413205, -315002, 407, 0, 1, 1, 14833, 260415373, 53414037505, 120965241901, 4744544613, 7156969, 1513, 1, 1

Offset: 0

Seiichi Manyama, May 20 2019

			For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + (-2)*(-x) + 4*(-x)^2/2 + (-8)*(-x)^3/6 + 14*(-x)^4/24 + (-20)*(-x)^5/120 + 20*(-x)^6/720. So A(3,2) = 1 - 2 + 4 - 8 + 14 - 20 + 20 = 9.
Square array begins:
   1, 1,   1,       1,            1,                  1, ...
   1, 0,   1,      -2,            9,                -44, ...
   1, 1,   3,      37,          997,              44121, ...
   1, 0,   9,    -692,       148041,          -66211704, ...
   1, 1,  31,   14371,     25413205,       120965241901, ...
   1, 0, 111, -315002,   4744544613,   -247578134832564, ...
   1, 1, 407, 7156969, 935728207597, 545591130328772081, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Columns k=0..5 give A000012, A059841, A120305, A307318, A307324, A308325.
Rows n=0..1 give A000012, A182386.
Main diagonal gives A308323.
Cf. A308292.

A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * (-x)^i/i! = (Sum_{i=0..n} x^i/i!)^k.

A371818 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n-2k,n-3*k).

Original entry on oeis.org

1, 2, 6, 19, 64, 224, 805, 2947, 10934, 40975, 154738, 587910, 2244681, 8605061, 33099767, 127687258, 493796454, 1913755319, 7431027611, 28902878561, 112585961052, 439148770623, 1715009647444, 6705019714554, 26240361155821, 102787164654287, 402972015656065

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A120305, A371819, A371820.

Cf. A000984, A144904.

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-2*k, n-3*k));

a(n) = [x^n] 1/((1-x+x^3) * (1-x)^n).

a(n) = binomial(2*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1/2-n, -n, 1+n], 27/4). - Stefano Spezia, Apr 07 2024

A371819 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n-k+1,n-3k).

Original entry on oeis.org

1, 3, 10, 34, 118, 417, 1497, 5447, 20047, 74493, 279054, 1052467, 3992204, 15216662, 58239175, 223688159, 861769598, 3328779906, 12887832493, 49998248601, 194315972151, 756406944446, 2948649839743, 11509316352548, 44976030493706, 175942932935325

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A001700, A120305, A371818, A371820.

Cf. A371773.

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-k+1, n-3*k));

a(n) = [x^n] 1/(((1-x)^2+x^3) * (1-x)^n).

a(n) = binomial(1+2*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [-1-2*n, 1+n/2, (3+n)/2], -27/4). - Stefano Spezia, Apr 07 2024

A371820 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n+2,n-3k).

Original entry on oeis.org

1, 4, 15, 55, 200, 726, 2640, 9636, 35343, 130339, 483395, 1802901, 6760781, 25482643, 96506229, 367077447, 1401772536, 5372120718, 20653929804, 79634421312, 307826528346, 1192608522258, 4629875048634, 18006340509702, 70142823370656, 273633773330844

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A001791, A120305, A371818, A371819.

Cf. A371777.

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n+2, n-3*k));

a(n) = [x^n] 1/(((1-x)^3+x^3) * (1-x)^n).
a(n) = binomial(2*(1+n), n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1+n/3, (4+n)/3, (5+n)/3], 1). - Stefano Spezia, Apr 07 2024
a(n) ~ 2^(2*n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Apr 19 2024

cp's OEIS Frontend

A307318 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).

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A307324 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (-1)^(i+j+k+l) * (i+j+k+l)!/(i!*j!*k!*l!).

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A307349 a(n) = Sum_{i=1..n} Sum_{j=1..n} (-1)^(i+j) * (i+j)!/(2!*i!*j!).

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A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

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A371798 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k-1,n-2*k).

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A306409 a(n) = -Sum_{0<=i

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A308322 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} (-1)^(i_1 + i_2 + ... + i_k) * multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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A371818 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-2*k,n-3*k).

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A307318 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).

A307324 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (-1)^(i+j+k+l) * (i+j+k+l)!/(i!j!k!*l!).

A307349 a(n) = Sum_{i=1..n} Sum_{j=1..n} (-1)^(i+j) * (i+j)!/(2!i!j!).

A371798 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-2k-1,n-2*k).

A371818 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n-2k,n-3*k).

A371819 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n-k+1,n-3k).

A371820 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2n+2,n-3k).