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A301305
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G.f. L(x,y) satisfies: L(x,y) = x * (1 + y*x*L'(x,y)) / (1 + x*L'(x,y)) where L'(x,y) = d/dx L(x,y), as a triangle read by rows.
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2
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1, -1, 1, 3, -5, 2, -14, 33, -25, 6, 85, -261, 292, -140, 24, -621, 2363, -3516, 2546, -892, 120, 5236, -23872, 44537, -43405, 23228, -6444, 720, -49680, 264860, -596396, 733983, -532095, 226644, -52356, 5040, 521721, -3193029, 8448004, -12605668, 11586756, -6707208, 2383248, -474144, 40320, -5994155, 41506739, -126480376, 222424796, -248535142, 182793154, -88379152, 27046632, -4745376, 362880, 74701055, -578419961, 1998774636, -4056699966, 5351696394, -4791391134, 2945757656, -1226765624, 330797184, -52079040, 3628800
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OFFSET
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1,4
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COMMENTS
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Main diagonal equals the factorials.
Rows sums are zeros after the initial row.
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LINKS
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FORMULA
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G.f. L(x,y) satisfies: [x^n] exp( -n * L(x,y) ) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp( -n * L(x,y) ) for n>=1.
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EXAMPLE
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G.f. L(x,y) = Sum_{n>=1, k=0..n-1} T(n,k) * x^n*y^k begins:
L(x,y) = x + (-1 + y)*x^2 + (3 - 5*y + 2*y^2)*x^3 + (-14 + 33*y - 25*y^2 + 6*y^3)*x^4 + (85 - 261*y + 292*y^2 - 140*y^3 + 24*y^4)*x^5 + (-621 + 2363*y - 3516*y^2 + 2546*y^3 - 892*y^4 + 120*y^5)*x^6 + (5236 - 23872*y + 44537*y^2 - 43405*y^3 + 23228*y^4 - 6444*y^5 + 720*y^6)*x^7 + (-49680 + 264860*y - 596396*y^2 + 733983*y^3 - 532095*y^4 + 226644*y^5 - 52356*y^6 + 5040*y^7)*x^8 + ...
where L = L(x,y) satisfies:
L = x*(1 + y*x*L') / (1 + x*L').
TRIANGLE.
This triangle of coefficients T(n,k) in L(x,y) begins:
1;
-1, 1;
3, -5, 2;
-14, 33, -25, 6;
85, -261, 292, -140, 24;
-621, 2363, -3516, 2546, -892, 120;
5236, -23872, 44537, -43405, 23228, -6444, 720;
-49680, 264860, -596396, 733983, -532095, 226644, -52356, 5040;
521721, -3193029, 8448004, -12605668, 11586756, -6707208, 2383248, -474144, 40320;
-5994155, 41506739, -126480376, 222424796, -248535142, 182793154, -88379152, 27046632, -4745376, 362880; ...
LIMITS.
In this triangle, the largest real root of the n-th row polynomial in y converges to the constant t = 2.845344903202547217277843362090557097661... (A301389).
RELATED SERIES.
exp(L(x,y)) = 1 + x + (-1 + 2*y)*x^2/2! + (13 - 24*y + 12*y^2)*x^3/3! + (-263 + 660*y - 540*y^2 + 144*y^3)*x^4/4! + (8381 - 26800*y + 31380*y^2 - 15840*y^3 + 2880*y^4)*x^5/5! + (-379409 + 1485870*y - 2280180*y^2 + 1706520*y^3 - 619200*y^4 + 86400*y^5)*x^6/6! + (22915369 - 106759128*y + 203726880*y^2 - 203269920*y^3 + 111449520*y^4 - 31691520*y^5 + 3628800*y^6)*x^7/7! + ...
satisfies: [x^n] exp(-n*L(x,y)) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp(-n*L(x,y)) for n>=1.
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PROG
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(PARI) {T(n, k) = my(L=x); for(i=0, n, L = x*(1 + y*x*L')/(1 + x*L' +x*O(x^n)) ); polcoeff(polcoeff(L, n, x), k, y)}
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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