A378610
Expansion of (1/x) * Series_Reversion( x * (1 - x/(1 - x))^4 ).
Original entry on oeis.org
1, 4, 30, 276, 2825, 30884, 353108, 4170500, 50485764, 623084056, 7810707894, 99175174284, 1272856327470, 16486135484248, 215212582153840, 2828658852385572, 37401956484705132, 497174193516767600, 6640063367021736728, 89058042321373540912, 1199031374607501831273
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x/(1-x))^4)/x)
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a(n, s=1, t=4, u=-4) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);
A378612
a(n) = Sum_{k=0..n} binomial(3*n+k-1,k) * binomial(n-1,n-k).
Original entry on oeis.org
1, 3, 27, 264, 2703, 28443, 304740, 3306852, 36225519, 399755001, 4437142467, 49485052224, 554059164036, 6224177431332, 70120015345512, 791898021185484, 8962485528377583, 101626868754849381, 1154295872365035537, 13130360954151723480, 149562006735075309783
Offset: 0
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a(n) = sum(k=0, n, binomial(3*n+k-1, k)*binomial(n-1, n-k));
A378611
a(n) = Sum_{k=0..n} binomial(2*n+k-1,k) * binomial(n-1,n-k).
Original entry on oeis.org
1, 2, 14, 104, 806, 6412, 51908, 425476, 3520070, 29332940, 245841284, 2070093632, 17499188924, 148414157816, 1262280506144, 10762045739644, 91951462167110, 787113739061260, 6749009521216052, 57954807274992208, 498334047795436276, 4290199618047230824
Offset: 0
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a(n) = sum(k=0, n, binomial(2*n+k-1, k)*binomial(n-1, n-k));
A378668
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)^2) )^2.
Original entry on oeis.org
1, 2, 13, 112, 1104, 11778, 132374, 1543740, 18505996, 226632616, 2823110349, 35659080952, 455652487060, 5879489288828, 76502741016012, 1002670573618324, 13224761472453756, 175404372357915096, 2338003752387818372, 31302169754776944512, 420760252068869028028
Offset: 0
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a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+2, n)/(4*n+k+2));
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a(n, r=2, s=1, t=5, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));
A378669
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^(4/3)/(1 - x*A(x)^(4/3)) )^3.
Original entry on oeis.org
1, 3, 21, 187, 1878, 20277, 229806, 2696523, 32478204, 399230972, 4988220669, 63165060093, 808828667104, 10455471983550, 136255868388684, 1788233397919211, 23614059664575324, 313531617379965156, 4183068478829324388, 56052027108881747724, 754020313029799707018
Offset: 0
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a(n) = 3*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+3, n)/(4*n+k+3));
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a(n, r=3, s=1, t=5, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));
Showing 1-5 of 5 results.