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A086754
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Pascal's square pyramid read by slices, each slice being read by rows. Each entry in slice n is the sum of the 4 entries above it in slice n-1.
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2
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1, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 3, 3, 1, 3, 9, 9, 3, 3, 9, 9, 3, 1, 3, 3, 1, 1, 4, 6, 4, 1, 4, 16, 24, 16, 4, 6, 24, 36, 24, 6, 4, 16, 24, 16, 4, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 5, 25, 50, 50, 25, 5, 10, 50, 100, 100, 50, 10, 10, 50, 100, 100, 50, 10, 5, 25, 50, 50, 25, 5, 1, 5, 10
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history;
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internal format)
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OFFSET
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1,7
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COMMENTS
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Element (i,j) of slice n is the coefficient of x^i * y^j in the expansion of ((1+x)*(1+y))^n. - Eitan Y. Levine, Sep 03 2023
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LINKS
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FORMULA
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C(n,i)*C(n,j) gives the (i,j) element in slice n, where C(n,k) are the binomial coefficients A007318.
G.f.: 1/(1-z(1+x)(1+y)) = Sum_{n>=0,i=0..n,j=0..n} T(n,i,j) * z^n * x^i * y^j
G.f. for slice n: ((1+x)*(1+y))^n = Sum_{i=0..n,j=0..n} T(n,i,j) * x^i * y^j (End)
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EXAMPLE
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The first 4 slices are
1..1 1..1 2 1..1 3 3 1
...1 1..2 4 2..3 9 9 3
........1 2 1..3 9 9 3
...............1 3 3 1
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MAPLE
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p:=n->seq(seq(binomial(n, i)*binomial(n, j), j=0..n), i=0..n): seq(p(n), n=0..5); # Emeric Deutsch, Nov 18 2004
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MATHEMATICA
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A[m_]:=Module[{pt=Table[ConstantArray[1, {i, i}], {i, m}]}, For[i=3, i<=m, i++, For[j=2, j<=i-1, j++, pt[[i, j, 1]]=pt[[i-1, j-1, 1]]+pt[[i-1, j, 1]]; pt[[i, 1, j]]=pt[[i, j, 1]]; pt[[i, i, j]]=pt[[i, j, 1]]; pt[[i, j, i]]=pt[[i, j, 1]]; ]; For[j=2, j<=i-1, j++, For[k=2, k<=i-1, k++, pt[[i, j, k]]=pt[[i-1, j, k]]+pt[[i-1, j, k-1]]+pt[[i-1, j-1, k]]+pt[[i-1, j-1, k-1]]; ]; ]; ]; pt//Flatten]; A[6] (* Robert P. P. McKone, Sep 14 2023, made from the PARI code *)
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PROG
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(PARI) { pt=vector(10, i, matrix(i, i, j, k, 1)); for (i=3, 10, for (j=2, i-1, pt[i][j, 1]=pt[i-1][j-1, 1]+pt[i-1][j, 1]; pt[i][1, j]=pt[i][j, 1]; pt[i][i, j]=pt[i][j, 1]; pt[i][j, i]=pt[i][j, 1]; ); for(j=2, i-1, for (k=2, i-1, pt[i][j, k]=pt[i-1][j, k]+pt[i-1][j, k-1]+pt[i-1][j-1, k]+pt[i-1][j-1, k-1]))); pt }
(Haskell)
a086754 n = a086754_list !! (n-1)
a086754_list = concat $ concat $ iterate ([[1, 1], [1, 1]] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
(p:ps) + (q:qs) = p + q : ps + qs
ps + qs = ps ++ qs
(p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
_ * _ = []
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CROSSREFS
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Consider the sequence s[i, j](n) obtained by considering the (i, j)-th entry of the n-th slice. Then if [i, j]= [3, 2] we get A006002, if [3, 3] we get A000537, if [4, 2] we get A004320, if [4, 3] we get A004282.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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