A375315
Expansion of (1 + x)/(1 - x^2*(1 + x)^3).
Original entry on oeis.org
1, 1, 1, 4, 7, 11, 23, 45, 81, 154, 296, 555, 1046, 1986, 3753, 7085, 13404, 25348, 47904, 90568, 171245, 323728, 612009, 1157071, 2187496, 4135527, 7818464, 14781237, 27944604, 52830706, 99879234, 188826693, 356986401, 674901117, 1275934888, 2412219633, 4560424135
Offset: 0
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my(N=40, x='x+O('x^N)); Vec((1+x)/(1-x^2*(1+x)^3))
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a(n) = sum(k=0, n\2, binomial(3*k+1, n-2*k));
A375317
Expansion of (1 + x)^2/(1 - x^2*(1 + x)^3).
Original entry on oeis.org
1, 2, 2, 5, 11, 18, 34, 68, 126, 235, 450, 851, 1601, 3032, 5739, 10838, 20489, 38752, 73252, 138472, 261813, 494973, 935737, 1769080, 3344567, 6323023, 11953991, 22599701, 42725841, 80775310, 152709940, 288705927, 545813094, 1031887518, 1950836005, 3688154521
Offset: 0
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CoefficientList[Series[(1+x)^2/(1-x^2(1+x)^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,3,3,1},{1,2,2,5,11},40] (* Harvey P. Dale, Mar 25 2025 *)
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my(N=40, x='x+O('x^N)); Vec((1+x)^2/(1-x^2*(1+x)^3))
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a(n) = sum(k=0, n\2, binomial(3*k+2, n-2*k));
A116089
Riordan array (1, x*(1+x)^3).
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 1, 15, 9, 1, 0, 0, 20, 36, 12, 1, 0, 0, 15, 84, 66, 15, 1, 0, 0, 6, 126, 220, 105, 18, 1, 0, 0, 1, 126, 495, 455, 153, 21, 1, 0, 0, 0, 84, 792, 1365, 816, 210, 24, 1, 0, 0, 0, 36, 924, 3003, 3060, 1330, 276, 27, 1
Offset: 0
Triangle begins as:
1;
0, 1;
0, 3, 1;
0, 3, 6, 1;
0, 1, 15, 9, 1;
0, 0, 20, 36, 12, 1;
0, 0, 15, 84, 66, 15, 1;
0, 0, 6, 126, 220, 105, 18, 1;
0, 0, 1, 126, 495, 455, 153, 21, 1;
0, 0, 0, 84, 792, 1365, 816, 210, 24, 1;
0, 0, 0, 36, 924, 3003, 3060, 1330, 276, 27, 1;
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Flat(List([0..12], n-> List([0..n], k-> Binomial(3*k, n-k) ))); # G. C. Greubel, May 09 2019
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[[Binomial(3*k, n-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 09 2019
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Flatten[Table[Binomial[3k,n-k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Feb 05 2012 *)
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{T(n,k) = binomial(3*k, n-k)}; \\ G. C. Greubel, May 09 2019
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[[binomial(3*k, n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 09 2019
A375314
a(n) = Sum_{k=0..floor(n/2)} binomial(4*k,n-2*k).
Original entry on oeis.org
1, 0, 1, 4, 7, 12, 30, 68, 137, 292, 644, 1380, 2936, 6324, 13625, 29216, 62701, 134784, 289547, 621708, 1335378, 2868620, 6161329, 13233352, 28424456, 61053608, 131135696, 281665480, 604991601, 1299461088, 2791106585, 5995016764, 12876698159, 27657841516
Offset: 0
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a(n) = sum(k=0, n\2, binomial(4*k, n-2*k));
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my(N=40, x='x+O('x^N)); Vec(1/(1-x^2*(1+x)^4))
A373741
Expansion of e.g.f. exp(x^2/2 * (1 + x)^3).
Original entry on oeis.org
1, 0, 1, 9, 39, 150, 1365, 13545, 105945, 918540, 10603845, 127806525, 1468823895, 18253765530, 257397445305, 3770163121725, 55637459903025, 866703333295800, 14468243658093225, 250223925107581425, 4426399346291497575, 81488489549760042750
Offset: 0
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With[{nn=30},CoefficientList[Series[Exp[x^2/2 (1+x)^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 26 2025 *)
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a(n) = n!*sum(k=0, n\2, binomial(3*k, n-2*k)/(2^k*k!));
A375307
a(n) = Sum_{k=0..floor(3*n/5)} binomial(3*n-3*k,2*k).
Original entry on oeis.org
1, 1, 4, 16, 52, 194, 685, 2452, 8771, 31327, 112004, 400285, 1430710, 5113647, 18277014, 65325542, 233485250, 834519021, 2982723523, 10660798289, 38103641048, 136189372297, 486765693153, 1739789499591, 6218325456983, 22225431015537, 79437750107600
Offset: 0
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a(n) = sum(k=0, 3*n\5, binomial(3*n-3*k, 2*k));
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my(N=30, x='x+O('x^N)); Vec((1-x-3*x^2)/(1-2*x-5*x^2-3*x^3+3*x^4-x^5))
Showing 1-6 of 6 results.