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Principal diagonal of the convolution array A213822.
+20
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4, 41, 147, 358, 710, 1239, 1981, 2972, 4248, 5845, 7799, 10146, 12922, 16163, 19905, 24184, 29036, 34497, 40603, 47390, 54894, 63151, 72197, 82068, 92800, 104429, 116991, 130522, 145058, 160635, 177289
FORMULA
a(n) = (-n - 3n^2 + 12*n^3)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: f(x)/g(x), where f(x) = x*(4 + 25*x + 7*x^2) and g(x) = (1-x)^4.
Antidiagonal sums of the convolution array A213822.
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4, 30, 114, 310, 690, 1344, 2380, 3924, 6120, 9130, 13134, 18330, 24934, 33180, 43320, 55624, 70380, 87894, 108490, 132510, 160314, 192280, 228804, 270300, 317200, 369954, 429030, 494914, 568110, 649140
FORMULA
a(n) = (2*n + 5*n^2 + 6*n^3 + 3*n^4)/4 = n*(1 + n)*(2 + 3*n + 3*n^2)/4.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: f(x)/g(x), where f(x) = 2*x*(2 + 5*x + 2*x^2) and g(x) = (1-x)^5.
Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.
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1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90
COMMENTS
Row 1: (1,2,3,...)**(1,2,3,...) = A000292. Row 2: (1,2,3,...)**(2,3,4,...) = A005581. Row 3: (1,2,3,...)**(3,4,5,...) = A006503. Row 4: (1,2,3,...)**(4,5,6,...) = A060488. Row 5: (1,2,3,...)**(5,6,7,...) = A096941. Row 6: (1,2,3,...)**(6,7,8,...) = A096957. ...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples: let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.
FORMULA
T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n: x*(n - (n - 1)*x)/(1 - x)^4.
EXAMPLE
Northwest corner (the array is read by southwest falling antidiagonals):
1, 4, 10, 20, 35, 56, 84, ...
2, 7, 16, 30, 50, 77, 112, ...
3, 10, 22, 40, 65, 98, 140, ...
4, 13, 28, 50, 80, 119, 168, ...
5, 16, 34, 60, 95, 140, 196, ...
6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.
MATHEMATICA
b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}] (* A213500 *)
PROG
(PARI)
t(n, k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k, n - k + 1), ", "); ); print(); ); };
(Python)
def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017
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