A261799
Number of 7-compositions of n: matrices with 7 rows of nonnegative integers with positive column sums and total element sum n.
Original entry on oeis.org
1, 7, 77, 819, 8687, 92141, 977347, 10366833, 109962202, 1166381804, 12371946734, 131230670312, 1391978902090, 14764881252772, 156612803600094, 1661210126351328, 17620647995924820, 186904251828901124, 1982515022137687464, 21028766197355391048
Offset: 0
-
a:= proc(n) option remember; `if`(n=0, 1,
add(a(n-j)*binomial(j+6, 6), j=1..n))
end:
seq(a(n), n=0..20);
A261800
Number of 8-compositions of n: matrices with 8 rows of nonnegative integers with positive column sums and total element sum n.
Original entry on oeis.org
1, 8, 100, 1208, 14554, 175352, 2112772, 25456328, 306717703, 3695574048, 44527157584, 536497912672, 6464145163032, 77885061063584, 938419943222768, 11306815168562400, 136233325153964242, 1641445323534504928, 19777413104380161776, 238293693669343744032
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..925
- Index entries for linear recurrences with constant coefficients, signature (16,-56,112,-140,112,-56,16,-2).
-
a:= proc(n) option remember; `if`(n=0, 1,
add(a(n-j)*binomial(j+7, 7), j=1..n))
end:
seq(a(n), n=0..20);
-
CoefficientList[Series[(1-x)^8/(2(1-x)^8-1),{x,0,30}],x] (* or *) LinearRecurrence[{16,-56,112,-140,112,-56,16,-2},{1,8,100,1208,14554,175352,2112772,25456328,306717703},30] (* Harvey P. Dale, Jul 15 2023 *)
A261801
Number of 9-compositions of n: matrices with 9 rows of nonnegative integers with positive column sums and total element sum n.
Original entry on oeis.org
1, 9, 126, 1704, 22986, 310086, 4183260, 56435004, 761346207, 10271072557, 138563678736, 1869317246556, 25218347263608, 340212470558832, 4589695110222504, 61918074814238448, 835316485437693186, 11268981358631127288, 152026139882340589466
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..880
- Index entries for linear recurrences with constant coefficients, signature (18, -72, 168, -252, 252, -168, 72, -18, 2).
-
a:= proc(n) option remember; `if`(n=0, 1,
add(a(n-j)*binomial(j+8, 8), j=1..n))
end:
seq(a(n), n=0..20);
A261802
Number of 10-compositions of n: matrices with 10 rows of nonnegative integers with positive column sums and total element sum n.
Original entry on oeis.org
1, 10, 155, 2320, 34640, 517252, 7723970, 115339960, 1722340115, 25719233330, 384058268507, 5735036957760, 85639736481880, 1278834734405320, 19096488909285540, 285162639746429024, 4258255614078447290, 63587365059302801520, 949532710487622388080
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..850
- Index entries for linear recurrences with constant coefficients, signature (20, -90, 240, -420, 504, -420, 240, -90, 20, -2).
-
a:= proc(n) option remember; `if`(n=0, 1,
add(a(n-j)*binomial(j+9, 9), j=1..n))
end:
seq(a(n), n=0..20);
A382924
Number of m-compositions of n with n zeros.
Original entry on oeis.org
1, 2, 13, 70, 336, 2076, 11091, 65210, 365661, 2159354, 11713047, 71427504, 392916687, 2245186352, 13527678851, 73679458270, 429472428457, 2553994191220, 14264421153074, 80483620074092, 489077890675807, 2768919905996888, 15394229582049408, 91794448088043258
Offset: 0
a(2) = 13 counts:
[2] [0] [0] [1] [1] [1] [0] [0] [0] [1][1] [1][0] [0][0] [0][1]
[0] [2] [0] [1] [0] [0] [1] [1] [0] [0][0], [0][1], [1][1], [1][0].
[0], [0], [2], [0] [1] [0] [1] [0] [1]
[0], [0], [1], [0], [1], [1],
-
G_tx(max_row) = {my(row = max_row, N = row*2, m = List([concat([1],vector(row-1,i,0))]), x='x+O('x^N), h=1 + sum(m=1,N,-1+ 1/(1 + t^m - (t + x/(1-x))^m))); for(n=1,row, listput(m,Vecrev(polcoeff(h, n))[1..row])); matrix(row, row, i,j, m[i][j])}
A382924(max_n) ={my(A=G_tx(max_n)); vector(max_n,i,A[i,i])}
A382924(20)
A383256
Number of n X n matrices of nonnegative entries with all columns summing to n and no horizontally adjacent zeros.
Original entry on oeis.org
1, 1, 7, 343, 125465, 366908001, 8698468668251, 1708834003295306868, 2810884261025802145414705, 39088555382409783097546399456477, 4626844513673581956954679383115038810744, 4688191496359773864437279635019555242588548880831
Offset: 0
a(1) = 1: [1]
a(2) = 7: [1,1] [1,0] [1,2] [0,1] [2,1] [0,2] [2,0]
[1,1], [1,2], [1,0], [2,1], [0,1], [2,0], [0,2].
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