A373928
Number of compositions of 7*n-2 into parts 1 and 7.
Original entry on oeis.org
1, 7, 35, 168, 819, 4025, 19796, 97315, 478304, 2350860, 11554621, 56791883, 279136551, 1371977475, 6743373646, 33144194898, 162906243014, 800696596250, 3935484773527, 19343207491818, 95073338508548, 467292702057555, 2296779231936167, 11288844908179562
Offset: 1
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a[n_]:= n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*HypergeometricPFQ[{1-n, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6, (10+n)/6}, {6/7, 8/7, 9/7, 10/7, 11/7, 12/7}, -6^6/7^7]/120; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
LinearRecurrence[{8,-21,35,-35,21,-7,1},{1,7,35,168,819,4025,19796},40] (* Harvey P. Dale, Jul 28 2024 *)
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a(n) = sum(k=0, n, binomial(n+4+6*k, n-1-k));
A373929
Number of compositions of 7*n-3 into parts 1 and 7.
Original entry on oeis.org
1, 6, 28, 133, 651, 3206, 15771, 77519, 380989, 1872556, 9203761, 45237262, 222344668, 1092840924, 5371396171, 26400821252, 129762048116, 637790353236, 3134788177277, 15407722718291, 75730131016730, 372219363549007, 1829486529878612, 8992065676243395
Offset: 1
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a[n_]:=n*(1 + n)*(2 + n)*(3 + n)*HypergeometricPFQ[{1-n, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6, (8+n)/6, (9+n)/6}, {5/7, 6/7, 8/7, 9/7, 10/7, 11/7}, -6^6/7^7]/24; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
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a(n) = sum(k=0, n, binomial(n+3+6*k, n-1-k));
A373931
Number of compositions of 7*n-5 into parts 1 and 7.
Original entry on oeis.org
1, 4, 17, 83, 413, 2037, 10010, 49183, 241722, 1188097, 5839638, 28702296, 141073905, 693388850, 3408058991, 16750869834, 82331801783, 404667078256, 1988969518921, 9775936716973, 48049473757425, 236166824233838, 1160777933797328, 5705311980035178
Offset: 1
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a[n_]:=n*(1 + n)*HypergeometricPFQ[{1-n,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, (7+n)/6}, {3/7, 4/7, 5/7, 6/7, 8/7, 9/7}, -6^6/7^7]/2; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
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a(n) = sum(k=0, n, binomial(n+1+6*k, n-1-k));
A373932
Number of compositions of 7*n-6 into parts 1 and 7.
Original entry on oeis.org
1, 3, 13, 66, 330, 1624, 7973, 39173, 192539, 946375, 4651541, 22862658, 112371609, 552314945, 2714670141, 13342810843, 65580931949, 322335276473, 1584302440665, 7786967198052, 38273537040452, 188117350476413, 924611109563490, 4544534046237850
Offset: 1
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a[n_]:=n*HypergeometricPFQ[{1-n,(1+n)/6,(2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6}, {2/7, 3/7, 4/7, 5/7, 6/7, 8/7}, -6^6/7^7]; Array[a,24] (* Stefano Spezia, Jun 23 2024 *)
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a(n) = sum(k=0, n, binomial(n+6*k, n-1-k));
Showing 1-4 of 4 results.