A277396
Coefficients in asymptotic expansion of sequence A277175.
Original entry on oeis.org
1, 1, 2, 7, 31, 163, 979, 6556, 48150, 383219, 3275121, 29841176, 288196506, 2936030427, 31425237185, 352166075233, 4119800015129, 50180781755797, 634948818421481, 8329111076372852, 113065244341635514, 1585699911447149109, 22942071009006046159
Offset: 0
A277175(n) / n! ~ 1 + 1/n + 2/n^2 + 7/n^3 + 31/n^4 + 163/n^5 + 979/n^6 + ...
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Flatten[{1, Table[Sum[CatalanNumber[j]*StirlingS2[n-1, j-1], {j, 1, n}], {n, 1, 25}]}]
A277359
Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1) and (1,0) on or below the diagonal and using steps (1,1), (-1,1), and (1,-1) on or above the diagonal.
Original entry on oeis.org
1, 2, 7, 32, 176, 1126, 8227, 67768, 622706, 6323932, 70400734, 852952952, 11176241098, 157506733030, 2375966883371, 38200984291800, 652179787654530, 11783182484950980, 224623760504277810, 4505795627243046240, 94873821120923655336, 2092249161797280567516
Offset: 0
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a:= proc(n) option remember; `if`(n<3, [1, 2, 7][n+1],
((n^3+10*n^2-10*n+1)*a(n-1)-(2*(4*n^3+2*n^2-29*n+28))
*a(n-2)+(4*(n-2))*(2*n-3)^2*a(n-3))/(n*(n+1)))
end:
seq(a(n), n=0..25);
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a[n_] := a[n] = If[n<3, {1, 2, 7}[[n+1]], ((n^3+10*n^2-10*n+1)*a[n-1] - (2*(4*n^3+2*n^2-29*n+28))*a[n-2] + (4*(n-2))*(2*n-3)^2*a[n-3])/(n*(n+1)) ]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 25 2017, translated from Maple *)
A277176
Exponential convolution of Catalan numbers and factorial numbers.
Original entry on oeis.org
1, 2, 6, 23, 106, 572, 3564, 25377, 204446, 1844876, 18465556, 203179902, 2438366836, 31699511768, 443795839192, 6656947282725, 106511191881270, 1810690391626380, 32592427526913540, 619256124778620450, 12385122502136529420, 260087572569333384840
Offset: 0
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a:= proc(n) option remember; `if`(n<2, n+1,
((n^2+5*n-2)*a(n-1)-(4*n-2)*(n-1)*a(n-2))/(n+1))
end:
seq(a(n), n=0..30);
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a[n_] := Sum[Binomial[n, i] CatalanNumber[i] (n-i)!, {i, 0, n}];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020 *)
Showing 1-3 of 3 results.
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