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Number of compositions of 7*n into parts 1 and 7.
+10
12
1, 2, 10, 53, 264, 1294, 6349, 31200, 153366, 753836, 3705166, 18211117, 89508951, 439943336, 2162355196, 10628140702, 52238121106, 256754344524, 1261967164192, 6202664757387, 30486569842400, 149843813435961, 736493759087077, 3619922936674360
FORMULA
a(n) = Sum_{k=0..n} binomial(n+6*k,n-k).
a(n) = 8*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x/(1 - x)^6).
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+6*k, n-k));
Number of compositions of 7*n into parts 6 and 7.
+10
12
1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 3448, 6556, 12121, 22509, 43453, 89150, 193823, 436304, 989759, 2219064, 4869285, 10434412, 21900170, 45297211, 93054446, 191371581, 396480142, 830227401, 1756883373, 3746468095, 8017653633, 17151612398
FORMULA
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^6/(1 - x)^6).
PROG
(PARI) a(n) = sum(k=0, n\6, binomial(n+k, n-6*k));
Number of compositions of 7*n into parts 2 and 7.
+10
6
1, 1, 2, 9, 38, 136, 452, 1495, 5031, 17114, 58282, 198032, 671856, 2278870, 7731892, 26238839, 89047335, 302191369, 1025487338, 3479970844, 11809261583, 40074827170, 135994407483, 461498426696, 1566098800484, 5314568565096, 18035031128780, 61202027710656
FORMULA
a(n) = Sum_{k=0..floor(n/2)} binomial(n+5*k,n-2*k).
a(n) = 7*a(n-1) - 20*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^2/(1 - x)^6).
PROG
(PARI) a(n) = sum(k=0, n\2, binomial(n+5*k, n-2*k));
Number of compositions of 7*n into parts 4 and 7.
+10
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));
Number of compositions of 7*n into parts 5 and 7.
+10
5
1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 794, 1732, 3553, 7116, 14501, 31078, 70607, 166922, 399315, 946121, 2197582, 4998597, 11188280, 24835641, 55117511, 123036293, 276976136, 628285812, 1431723937, 3265884047, 7436635822, 16880558594, 38196652951, 86238054374
FORMULA
a(n) = Sum_{k=0..floor(n/5)} binomial(n+2*k,n-5*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 22*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: 1/(1 - x - x^5/(1 - x)^6).
MATHEMATICA
LinearRecurrence[{7, -21, 35, -35, 22, -7, 1}, {1, 1, 1, 1, 1, 2, 9}, 40] (* Harvey P. Dale, Oct 19 2024 *)
PROG
(PARI) a(n) = sum(k=0, n\5, binomial(n+2*k, n-5*k));
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