A163773
Row sums of the swinging derangement triangle (A163770).
Original entry on oeis.org
1, 1, 4, 15, -14, 185, -454, 2107, -6194, 22689, -70058, 234971, -734304, 2368379, -7404318, 23417955, -72988938, 228324569, -708982738, 2202742447, -6815736144, 21077285943, -65016664062, 200371842727, -616463969324, 1894794918275, -5816606133674, 17839764136377
Offset: 0
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swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i,k; add(add((-1)^(n-i)*binomial(n-k,n-i)*swing(i),i=k..n), k=0..n) end:
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sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 03 2017 *)
A163771
Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).
Original entry on oeis.org
1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870
Offset: 0
Triangle begins
1;
1, 2;
3, 4, 6;
7, 10, 14, 20;
19, 26, 36, 50, 70;
51, 70, 96, 132, 182, 252;
141, 192, 262, 358, 490, 672, 924;
From _M. F. Hasler_, Nov 15 2019: (Start)
The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:
1 1 3 7 19 51 ...
2 4 10 26 70 192 ...
6 14 36 96 262 726 ...
20 50 132 358 988 2760 ...
70 182 490 1346 3748 10540 ...
252 672 1836 5094 14288 40404 ...
(...)
Read by falling antidiagonals this yields the same sequence. (End)
-
For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k),n,true);
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sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
A163772
Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse. Triangle read by rows. For n >= 0, k >= 0.
Original entry on oeis.org
1, 5, 6, 19, 24, 30, 67, 86, 110, 140, 227, 294, 380, 490, 630, 751, 978, 1272, 1652, 2142, 2772, 2445, 3196, 4174, 5446, 7098, 9240, 12012, 7869, 10314, 13510, 17684, 23130, 30228, 39468, 51480
Offset: 0
Triangle begins:
1;
5, 6;
19, 24, 30;
67, 86, 110, 140;
227, 294, 380, 490, 630;
751, 978, 1272, 1652, 2142, 2772;
2445, 3196, 4174, 5446, 7098, 9240, 12012;
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For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k+1),n,true);
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sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[ (-1)^(n-i)*Binomial[n-k, n-i]*sf[2*i+1], {i, k, n}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Showing 1-3 of 3 results.
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