Search: a351776 -id:a351776
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A351761
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.
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6
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1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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E.g.f. of column k: 1/(1 - k*x*exp(x)).
T(0,k) = 1 and T(n,k) = k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 4, 12, 24, 40, 60, ...
0, 21, 102, 279, 588, 1065, ...
0, 148, 1160, 4332, 11536, 25220, ...
0, 1305, 16490, 84075, 282900, 746525, ...
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PROG
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(PARI) T(n, k) = n!*sum(j=0, n, k^(n-j)*(n-j)^j/j!);
(PARI) T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
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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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A351777
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Expansion of e.g.f. 1/(1 + 2*x*exp(x)).
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2
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1, -2, 4, -6, -8, 150, -972, 3682, 6256, -289746, 3300460, -21071622, -27876312, 3156947014, -53217341660, 494232431250, 175171749088, -113735274256290, 2613309376750812, -32653995355358678, 36013529538641560, 10227377502146048118, -305630239215263764076
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * (n-k)^k/k!.
a(0) = 1 and a(n) = -2 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
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MATHEMATICA
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With[{nn=30}, CoefficientList[Series[1/(1+2x Exp[x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Feb 06 2024 *)
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PROG
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(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+2*x*exp(x))))
(PARI) a(n) = n!*sum(k=0, n, (-2)^(n-k)*(n-k)^k/k!);
(PARI) a(n) = if(n==0, 1, -2*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A351778
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Expansion of e.g.f. 1/(1 + 3*x*exp(x)).
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3
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1, -3, 12, -63, 420, -3435, 33462, -379155, 4903896, -71318259, 1152202290, -20474486043, 396890715636, -8334602179995, 188486823883134, -4567087352339235, 118039115079323952, -3241465018561379427, 94249758656850366186, -2892678859033260044043
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * (n-k)^k/k!.
a(0) = 1 and a(n) = -3 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
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MATHEMATICA
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With[{nn=30}, CoefficientList[Series[1/(1+3x Exp[x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 21 2024 *)
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PROG
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(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*x*exp(x))))
(PARI) a(n) = n!*sum(k=0, n, (-3)^(n-k)*(n-k)^k/k!);
(PARI) a(n) = if(n==0, 1, -3*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A351779
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a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * (n-k)^k/k!.
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3
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1, -1, 4, -63, 2288, -138525, 12381084, -1528482823, 249005711296, -51739455340953, 13353206066063900, -4190486732316600771, 1571373340568392914288, -693899460077821703051125, 356404409990391961980227068, -210670220153918100996704166975
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = n! * [x^n] 1/(1 + n*x*exp(x)).
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PROG
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(PARI) a(n) = n!*sum(k=0, n, (-n)^(n-k)*(n-k)^k/k!);
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A351791
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k * (n-j))^j/j!.
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+0
3
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1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, -6, -4, 120, 1, 1, -6, -3, 40, 25, 720, 1, 1, -8, 6, 132, 120, 114, 5040, 1, 1, -10, 21, 248, -375, -1872, -287, 40320, 1, 1, -12, 42, 364, -2120, -8298, -3920, -4152, 362880, 1, 1, -14, 69, 456, -5655, -12144, 86121, 155776, -1647, 3628800
(list;
table;
graph;
refs;
listen;
history;
text;
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OFFSET
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0,6
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LINKS
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FORMULA
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E.g.f. of column k: 1/(1 - x*exp(-k*x)).
T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 0, -2, -4, -6, -8, ...
6, -3, -6, -3, 6, 21, ...
24, -4, 40, 132, 248, 364, ...
120, 25, 120, -375, -2120, -5655, ...
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MATHEMATICA
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T[n_, k_] := n!*(1 + Sum[(-k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
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PROG
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(PARI) T(n, k) = n!*sum(j=0, n, (-k*(n-j))^j/j!);
(PARI) T(n, k) = if(n==0, 1, n*sum(j=0, n-1, (-k)^(n-1-j)*binomial(n-1, j)*T(j, k)));
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CROSSREFS
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Main diagonal gives (-1)^n * A302398(n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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