A145515
Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 10, 1, 1, 1, 2, 6, 23, 36, 1, 1, 1, 2, 7, 46, 239, 202, 1, 1, 1, 2, 8, 82, 1086, 5828, 1828, 1, 1, 1, 2, 9, 134, 3707, 79326, 342383, 27338, 1, 1, 1, 2, 10, 205, 10340, 642457, 18583582, 50110484, 692004, 1, 1, 1, 2, 11, 298, 24901, 3649346, 446020582, 14481808030, 18757984046, 30251722, 1, 1
Offset: 0
A(2,3) = 5, because there are 5 partitions of 3^2=9 into powers of 3: [1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,3], [1,1,1,3,3], [3,3,3], [9].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 2, 2, 2, ...
1, 1, 4, 5, 6, 7, ...
1, 1, 10, 23, 46, 82, ...
1, 1, 36, 239, 1086, 3707, ...
1, 1, 202, 5828, 79326, 642457, ...
Columns k=0+1, 2-10 give:
A000012,
A002577,
A078125,
A078537,
A111822,
A111827,
A111832,
A111837,
A145512,
A145513.
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b:= proc(n, j, k) local nn;
nn:= n+1;
if n<0 then 0
elif j=0 or n=0 or k<=1 then 1
elif j=1 then nn
elif n>=j then (nn-j) *binomial(nn, j) *add(binomial(j, h)
/(nn-j+h) *b(j-h-1, j, k) *(-1)^h, h=0..j-1)
else b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)
fi
end:
A:= (n, k)-> b(1, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..13);
-
b[n_, j_, k_] := Module[{nn = n+1}, Which[n < 0, 0, j == 0 || n == 0 || k <= 1, 1, j == 1, nn, n >= j, (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)* b[j-h-1, j, k]*(-1)^h, {h, 0, j-1}], True, b[n, j, k] = b[n-1, j, k] + b[k*n, j-1, k] ] ]; a[n_, k_] := b[1, n, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
A111830
Triangle P, read by rows, that satisfies [P^7](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(7*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(0,k)=1 and P(k,k)=1 for all k>=0.
Original entry on oeis.org
1, 1, 1, 1, 7, 1, 1, 154, 49, 1, 1, 16275, 8281, 343, 1, 1, 9106461, 6558209, 410914, 2401, 1, 1, 28543862991, 27307109501, 2298650515, 20170801, 16807, 1, 1, 521136519414483, 636922972420469, 67522139062441, 790856748801, 988621354
Offset: 0
Let q=7; the g.f. of column k of matrix power P^m is:
1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +
(m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +
(m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...
where L(x) satisfies:
x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...
and L(x) = x - 5/2!*x^2 + 83/3!*x^3 + 16110/4!*x^4 +... (A111834).
Thus the g.f. of column 0 of matrix power P^m is:
1 + m*L(x) + m^2/2!*L(x)*L(7*x) + m^3/3!*L(x)*L(7*x)*L(7^2*x) +
m^4/4!*L(x)*L(7*x)*L(7^2*x)*L(7^3*x) + ...
Triangle P begins:
1;
1,1;
1,7,1;
1,154,49,1;
1,16275,8281,343,1;
1,9106461,6558209,410914,2401,1;
1,28543862991,27307109501,2298650515,20170801,16807,1; ...
where P^7 shifts columns left and up one place:
1;
7,1;
154,49,1;
16275,8281,343,1; ...
A111822
Number of partitions of 5^n into powers of 5, also equals the row sums of triangle A111820, which shifts columns left and up under matrix 5th power.
Original entry on oeis.org
1, 2, 7, 82, 3707, 642457, 446020582, 1288155051832, 15905066118254957, 856874264098480364332, 204616369654716156089739332, 219286214391142987407272329973707, 1065403165201779499307991460987124895582, 23663347632778954225192551079067428619449114332
Offset: 0
-
a(n,q=5)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(sum(k=0,n,A[n+1,k+1])))
A111827
Number of partitions of 6^n into powers of 6, also equals the row sums of triangle A111825, which shifts columns left and up under matrix 6th power.
Original entry on oeis.org
1, 2, 8, 134, 10340, 3649346, 6188114528, 52398157106366, 2277627698797283420, 518758596372421679994170, 628925760908337480420110203736, 4109478867142143642923124190955500214
Offset: 0
-
a(n,q=6)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(sum(k=0,n,A[n+1,k+1])))
A111837
Number of partitions of 8^n into powers of 8, also equals the row sums of triangle A111835, which shifts columns left and up under matrix 8th power.
Original entry on oeis.org
1, 2, 10, 298, 53674, 58573738, 409251498922, 19046062579215274, 6071277235712979102634, 13531779463193107731083553706, 214224474679766323250278564215516074, 24390479071277895100812271376578637910371242, 20173309182842708837666031701435147789403500172143530
Offset: 0
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a(n,q=8)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(sum(k=0,n,A[n+1,k+1])))
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