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A253381
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Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+2k)^k.
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3
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1, -3, 2, -3, -22, 3, -3, 122, -69, 4, -3, -518, 891, -156, 5, -3, 1882, -8709, 3444, -295, 6, -3, -6182, 71931, -57036, 9785, -498, 7, -3, 18906, -530181, 789684, -241095, 23022, -777, 8, -3, -54822, 3598587, -9661260, 4919865, -783378, 47607, -1144, 9, -3, 152538, -22943493, 107911860, -87977415, 21896622, -2129673, 89576, -1611, 10
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OFFSET
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0,2
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COMMENTS
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Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^0 + T(n,1)*(x+2)^1 + T(n,2)*(x+4)^2 + ... + T(n,n)*(x+2n)^n for n >= 0.
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LINKS
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FORMULA
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T(n,n) = n+1, for n >= 0.
T(n,n-1) = n*(1 - 2*n - 2*n^2), for n >= 1.
T(n,n-2) = (n-1)*(2*n^4-2*n^3-6*n^2+2*n+1), for n >= 2.
T(n,n-3) = (2-n)*(4*n^6-24*n^5+26*n^4+54*n^3-72*n^2+9)/3, for n >= 3.
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EXAMPLE
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The triangle T(n,k) starts:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: -3 2
2: -3 -22 3
3: -3 122 -69 4
4: -3 -518 891 -156 5
5: -3 1882 -8709 3444 -295 6
6: -3 -6182 71931 -57036 9785 -498 7
7: -3 18906 -530181 789684 -241095 23022 -777 8
8: -3 -54822 3598587 -9661260 4919865 -783378 47607 -1144 9
9 : -3 152538 -22943493 107911860 -87977415 21896622 -2129673 89576 -1611 10
... Reformatted.
----------------------------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = -3*(x+0)^0 + 122*(x+2)^1 - 69*(x+4)^2 + 4* (x+6)^3. (End)
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PROG
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(PARI) T(n, k) = (k+1)-sum(i=k+1, n, (2*i)^(i-k)*binomial(i, k)*T(n, i))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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