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A128570
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Rectangular table, read by antidiagonals, where the g.f. of row n, R(x,n), satisfies: R(x,n) = 1 + (n+1)*x*R(x,n+1)^2 for n>=0.
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11
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1, 1, 1, 1, 2, 4, 1, 3, 12, 28, 1, 4, 24, 114, 276, 1, 5, 40, 288, 1440, 3480, 1, 6, 60, 580, 4440, 22368, 53232, 1, 7, 84, 1020, 10560, 82080, 409248, 955524, 1, 8, 112, 1638, 21420, 226560, 1752000, 8585088, 19672320, 1, 9, 144, 2464, 38976, 523320, 5532960, 42178800, 202733760, 456803328, 1, 10, 180, 3528, 65520, 1068480, 14399280, 150570240, 1127335680, 5317663680, 11810032896, 1, 11, 220, 4860, 103680, 1991808, 32716992, 437433780, 4501422240, 33073099200, 153345634560, 336463895808
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refs;
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OFFSET
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0,5
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COMMENTS
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LINKS
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EXAMPLE
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Row g.f.s satisfy: R(x,0) = 1 + x*R(x,1)^2, R(x,1) = 1 + 2x*R(x,2)^2,
R(x,2) = 1 + 3x*R(x,3)^2, R(x,3) = 1 + 4x*R(x,4)^2, ...
where the initial rows begin:
R(x,0):[1,1,4,28,276,3480,53232,955524,19672320,456803328,...];
R(x,1):[1,2,12,114,1440,22368,409248,8585088,202733760,...];
R(x,2):[1,3,24,288,4440,82080,1752000,42178800,1127335680,...];
R(x,3):[1,4,40,580,10560,226560,5532960,150570240,4501422240,...];
R(x,4):[1,5,60,1020,21420,523320,14399280,437433780,14479664640,...];
R(x,5):[1,6,84,1638,38976,1068480,32716992,1098069504,39896236800,...];
R(x,6):[1,7,112,2464,65520,1991808,67189248,2469837888,97765355520,..];
R(x,7):[1,8,144,3528,103680,3461760,127569600,5098406400,...];
R(x,8):[1,9,180,4860,156420,5690520,227470320,9821970180,...];
R(x,9):[1,10,220,6490,227040,8939040,385265760,17875608960,..].
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PROG
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(PARI) {T(n, k)=local(A=1+(n+k+1)*x); for(j=0, k, A=1+(n+k+1-j)*x*A^2 +x*O(x^k)); polcoeff(A, k)}
for(n=0, 12, for(k=0, 10, 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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