A351725 -id:A351725 - cp's OEIS frontend

A351726 Table T(n,k) read by rows: number of compositions of n into k parts of size 1, 5, 10 or 25.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 6, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 3, 6, 0, 0, 0, 8, 0, 0, 0, 1, 0, 0, 0, 0, 4

Offset: 0

R. J. Mathar, Feb 17 2022

			T(7,3)=3 counts 1+1+5 =1+5+1 =5+1+1.
T(10,2)=1 counts 5+5.
T(12,3)=3 counts 1+1+10 =1+10+1 =10+1+1.
T(15,3)=1 counts 5+5+5.
T(16,3)=6 counts 1+5+10 =1+10+5 =5+1+10 =5+10+1 =10+1+5 =10+5+1.
The triangle starts in row n=0 and columns 0<=k<=n:
1
0 1
0 0 1
0 0 0 1
0 0 0 0  1
0 1 0 0  0  1
0 0 2 0  0  0  1
0 0 0 3  0  0  0  1
0 0 0 0  4  0  0  0  1
0 0 0 0  0  5  0  0  0  1
0 1 1 0  0  0  6  0  0  0   1
0 0 2 3  0  0  0  7  0  0   0  1
0 0 0 3  6  0  0  0  8  0   0  0  1
0 0 0 0  4 10  0  0  0  9   0  0  0  1
0 0 0 0  0  5 15  0  0  0  10  0  0  0  1
0 0 2 1  0  0  6 21  0  0   0 11  0  0  0  1
0 0 0 6  4  0  0  7 28  0   0  0 12  0  0  0  1
0 0 0 0 12 10  0  0  8 36   0  0  0 13  0  0  0  1
0 0 0 0  0 20 20  0  0  9  45  0  0  0 14  0  0  0 1
0 0 0 0  0  0 30 35  0  0  10 55  0  0  0 15  0  0 0 1
0 0 1 3  1  0  0 42 56  0   0 11 66  0  0  0 16  0 0 0 1
0 0 0 3 12  5  0  0 56 84   0  0 12 78  0  0  0 17 0 0 0 1
0 0 0 0  6 30 15  0  0 72 120  0  0 13 91  0  0  0 18 0 0 0 1

L. Zhiwen, Expect coin value problem (a variant of coin exchange problem), math.stackexchange Feb 12, 2022.
Index entries for sequences related to making change.

Cf. A351724 (row sums), A351725 (partitions).

T(n,0) = 0 if k>0.

G.f.: 1/(1-y*g(x)) where g(x)=x+x^5+x^10+x^25 is the g.f. of column k=1.

A351740 Total number of parts in all partitions of n into parts of size 1, 5, 10 or 25.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 19, 23, 27, 31, 35, 44, 50, 56, 62, 68, 83, 92, 101, 110, 119, 141, 154, 167, 180, 193, 226, 244, 262, 280, 298, 343, 367, 391, 415, 439, 500, 531, 562, 593, 624, 702, 741, 780, 819, 858, 959, 1008, 1057, 1106, 1155, 1280, 1340

Offset: 0

Alois P. Heinz, Feb 17 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001299, A351725.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
     (p->`if`(p>n, 0, (t->t+[0, t[1]])(b(n-p, i))))([1, 5, 10, 25][i]))
    end:
a:= n-> b(n, 4)[2]:
seq(a(n), n=0..100);

a(n) = Sum_{k=0..n} k * A351725(n,k).

A351742 Number of partitions of 2n into n parts of size 1, 5, 10 or 25.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 3, 2, 2, 1, 3, 3, 2, 1, 4, 3, 2, 2, 4, 3, 3, 2, 4, 4, 3, 2, 5, 4, 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 7, 5, 4, 4, 7, 5, 5, 4, 7, 7, 5, 4, 8, 7, 5

Offset: 0

Alois P. Heinz, Feb 17 2022

			a(0) = 1: [].
a(4) = 1: [{5}^1,{1}^3].
a(24) = 2: [{5}^6,{1}^18], [{25}^1,{1}^23].
a(36) = 3: [{5}^9,{1}^27], [{10}^4,{1}^32], [{25}^1,{5}^3,{1}^32].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,-1,0,0,0,1).

Cf. A351725.

G.f.: -1/(x^37-x^33-x^28+x^24-x^13+x^9+x^4-1).

a(n) = A351725(2n,n).

cp's OEIS Frontend

A351726 Table T(n,k) read by rows: number of compositions of n into k parts of size 1, 5, 10 or 25.

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A351740 Total number of parts in all partitions of n into parts of size 1, 5, 10 or 25.

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A351742 Number of partitions of 2n into n parts of size 1, 5, 10 or 25.

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