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Expansion of 1/(1 - x^4/(1-x)^7).
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6
1, 0, 0, 0, 1, 7, 28, 84, 211, 476, 1029, 2276, 5384, 13594, 35371, 91667, 232681, 577710, 1413462, 3442498, 8414484, 20717963, 51346109, 127678961, 317496621, 787941379, 1950774874, 4821609252, 11910608942, 29432604429, 72787392898, 180131835001
COMMENTS
Number of compositions of 7*n-4 into parts 4 and 7.
FORMULA
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 34*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 7.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-1+3*k,n-4*k).
PROG
(PARI) my(N=40, x='x+O('x^N)); Vec(1/(1-x^4/(1-x)^7))
(PARI) a(n) = sum(k=0, n\4, binomial(n-1+3*k, n-4*k));
Expansion of 1/(1 - x^2 - x^7).
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5
1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 4, 6, 7, 7, 11, 9, 16, 13, 22, 20, 29, 31, 38, 47, 51, 69, 71, 98, 102, 136, 149, 187, 218, 258, 316, 360, 452, 509, 639, 727, 897, 1043, 1257, 1495, 1766, 2134, 2493, 3031, 3536, 4288, 5031, 6054, 7165, 8547, 10196, 12083, 14484, 17114, 20538, 24279, 29085, 34475
COMMENTS
Number of compositions of n into parts 2 and 7.
PROG
(PARI) my(N=70, x='x+O('x^N)); Vec(1/(1-x^2-x^7))
(PARI) a(n) = sum(k=0, n\7, ((n-5*k)%2==0)*binomial((n-5*k)/2, k));
Expansion of 1/(1 - x^3 - x^7).
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5
1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 1, 1, 4, 3, 1, 5, 6, 2, 6, 10, 5, 7, 15, 11, 9, 21, 21, 14, 28, 36, 25, 37, 57, 46, 51, 85, 82, 76, 122, 139, 122, 173, 224, 204, 249, 346, 343, 371, 519, 567, 575, 768, 913, 918, 1139, 1432, 1485, 1714, 2200, 2398, 2632, 3339, 3830, 4117, 5053, 6030, 6515, 7685
COMMENTS
Number of compositions of n into parts 3 and 7.
PROG
(PARI) my(N=80, x='x+O('x^N)); Vec(1/(1-x^3-x^7))
(PARI) a(n) = sum(k=0, n\7, ((n-4*k)%3==0)*binomial((n-4*k)/3, k));
Expansion of 1/(1 - x^5 - x^7).
+10
5
1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 1, 1, 4, 0, 6, 1, 4, 5, 1, 10, 1, 10, 6, 5, 15, 2, 20, 7, 15, 21, 7, 35, 9, 35, 28, 22, 56, 16, 70, 37, 57, 84, 38, 126, 53, 127, 121, 95, 210, 91, 253, 174, 222, 331, 186, 463, 265, 475, 505, 408, 794, 451, 938, 770, 883, 1299, 859, 1732, 1221
COMMENTS
Number of compositions of n into parts 5 and 7.
FORMULA
a(n) = a(n-5) + a(n-7).
Gf.: 1/((1-x+x^2)*(1+x-x^3-x^4-x^5)) . - R. J. Mathar, Jul 03 2024
PROG
(PARI) my(N=80, x='x+O('x^N)); Vec(1/(1-x^5-x^7))
(PARI) a(n) = sum(k=0, n\7, ((n-2*k)%5==0)*binomial((n-2*k)/5, k));
Number of compositions of 7*n into parts 4 and 7.
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5
1, 1, 1, 1, 2, 9, 37, 121, 332, 808, 1837, 4113, 9497, 23091, 58462, 150129, 382810, 960520, 2373982, 5816480, 14230964, 34948927, 86295036, 213973997, 531470618, 1319411997, 3270186871, 8091796123, 20002405065, 49435009494, 122222402392, 302354237393
FORMULA
a(n) = Sum_{k=0..floor(n/4)} binomial(n+3*k,n-4*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 34*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^4/(1 - x)^6).
PROG
(PARI) a(n) = sum(k=0, n\4, binomial(n+3*k, n-4*k));
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