A263070
Number of lattice paths from {n}^10 to {0}^10 using steps that decrement one or more components by one.
Original entry on oeis.org
1, 102247563, 1843200116875263613, 93777824804632275267836362863, 7581761490297442738124283591348762605121, 797656368265147949572521540584234236944835806750363, 99479717242433942914309980793245660313479486272546475327799069
Offset: 0
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With[{k = 10}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 10}]] (* Vaclav Kotesovec, Mar 22 2016 *)
A263071
Number of lattice paths from {9}^n to {0}^n using steps that decrement one or more components by one.
Original entry on oeis.org
1, 1, 1462563, 191731486403293, 496505991344667030490635, 12024609569670508078686022988554381, 1742079663955078309800553960117733249663480043, 1121241285685659360225420876424590015281785102622410968973
Offset: 0
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With[{r = 9}, Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 10}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)
A263072
Number of lattice paths from {10}^n to {0}^n using steps that decrement one or more components by one.
Original entry on oeis.org
1, 1, 8097453, 9850349744182729, 331910222316215755702672557, 134565509066155510620216211257550349401, 399017534874989738901076297624977315332337599285373, 6213239693876579408708842528154872834110410698303331900339282569
Offset: 0
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With[{r = 10}, Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 10}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)
A331637
Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones.
Original entry on oeis.org
1, 4, 14, 89, 542, 5156, 47294, 570074, 7103343, 104493608, 1622632574, 28411638982, 526858348382, 10699857854372, 230297862127828, 5330284144906443, 130370767029135902, 3390229775073259800, 92801587319328411134, 2679531116538955252338
Offset: 1
The a(2) = 4 matrices are:
[1] [1 0] [0 1] [1 1]
[1] [0 1] [1 0]
A384351
Expansion of Product_{k>=1} 1/(1 - k*(k+1)/2 * x)^((1/2)^(k+2)).
Original entry on oeis.org
1, 1, 7, 143, 6140, 455828, 51947988, 8414718996, 1836791273514, 519582028795210, 184852108308617398, 80776494267416227078, 42529172631705836804876, 26553065315757661351020284, 19397441882229095276127402500, 16390942374821715002096327774628
Offset: 0
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nmax = 20; CoefficientList[Series[Exp[Sum[Sum[Sum[(-1)^j*Binomial[i, j]*((i - j)*(i - j - 1)/2)^k, {j, 0, i}], {i, 0, 2 k}]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 29 2025 *)
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a262809(n, k) = sum(i=0, k*n, sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a262809(2, k)*x^k/k)))
A384352
Expansion of Product_{k>=1} 1/(1 - k*(k+1)*(k+2)/6 * x)^((1/2)^(k+3)).
Original entry on oeis.org
1, 1, 32, 5392, 2676188, 2930633692, 5993325199448, 20540879727692152, 109337218761743017718, 854254522610491562826582, 9378640254148405369808277352, 139752461092050444767050922501096, 2747716352285121538660626991038190636, 69628008338488529846443753577404293410060
Offset: 0
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a262809(n, k) = sum(i=0, k*n, sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a262809(3, k)*x^k/k)))
A384353
Expansion of Product_{k>=1} 1/(1 - k*(k+1)*(k+2)*(k+3)/24 * x)^((1/2)^(k+4)).
Original entry on oeis.org
1, 1, 161, 233201, 1388333781, 23407417517205, 900363695229160325, 68584682130559722233525, 9362104205577409136806214275, 2125938144923623062958782871506275, 758178276483321320080629434392636915075, 405630344408921348237973282862682052175313075
Offset: 0
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a262809(n, k) = sum(i=0, k*n, sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a262809(4, k)*x^k/k)))
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