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A142472
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Triangle T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k), read by rows.
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1
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1, -4, 1, 21, -18, 1, -140, 240, -48, 1, 1140, -3150, 1300, -100, 1, -11004, 43620, -29700, 4800, -180, 1, 123074, -650769, 647780, -175175, 13965, -294, 1, -1566928, 10517108, -14190400, 5676160, -764400, 34496, -448, 1, 22390488, -184052520, 319680732, -175091112, 35160048, -2698920, 75600, -648, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Row sums are: 1, -3, 4, 53, -809, 7537, -41418, -294411, 15463669, -352665269, ....
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LINKS
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G. C. Greubel, Rows n = 1..50 of the triangle, flattened
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FORMULA
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T(n, k) = binomial(n, k) * Sum_{j=k..n} StirlingS1(n, j)*StirlingS1(j, k).
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EXAMPLE
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The triangle begins as:
1;
-4, 1;
21, -18, 1;
-140, 240, -48, 1;
1140, -3150, 1300, -100, 1;
-11004, 43620, -29700, 4800, -180, 1;
123074, -650769, 647780, -175175, 13965, -294, 1;
-1566928, 10517108, -14190400, 5676160, -764400, 34496, -448, 1;
22390488, -184052520, 319680732, -175091112, 35160048, -2698920, 75600, -648, 1;
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MAPLE
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A142472:= (n, k)-> binomial(n, k)*add(Stirling1(n, j)*Stirling1(j, k), j=k..n);
seq(seq(A142472(n, k), k=1..n), n=1..12); # G. C. Greubel, Apr 02 2021
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MATHEMATICA
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T[n_, k_]:= Binomial[n, k]*Sum[StirlingS1[n, j]*StirlingS1[j, k], {j, k, n}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 02 2021 *)
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PROG
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(Magma)
A142472:= func< n, k | Binomial(n, k)*(&+[StirlingFirst(n, j)*StirlingFirst(j, k): j in [k..n]]) >;
[A142472(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 02 2021
(Sage)
def A142472(n, k): return (-1)^(n-k)*binomial(n, k)*sum( stirling_number1(n, j)*stirling_number1(j, k) for j in (k..n) )
flatten([[A142472(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 02 2021
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CROSSREFS
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Cf. A008275, A039814.
Sequence in context: A126457 A159841 A202550 * A360089 A299445 A135049
Adjacent sequences: A142469 A142470 A142471 * A142473 A142474 A142475
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KEYWORD
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sign,tabl
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AUTHOR
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Roger L. Bagula and Gary W. Adamson, Sep 22 2008
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EXTENSIONS
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Edited by N. J. A. Sloane, Sep 26 2008
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STATUS
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approved
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