Revision History for A154948
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#12 by Charles R Greathouse IV at Thu Sep 08 08:45:40 EDT 2022
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(MAGMAMagma) [ (&+[Binomial(k-1, j)*Binomial(n-j+1, k+1): j in [0..n+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020
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Discussion
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| Thu Sep 08
| 08:45
| OEIS Server: https://oeis.org/edit/global/2944
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#11 by OEIS Server at Tue Feb 18 20:28:30 EST 2020
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| LINKS
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G. C. Greubel, <a href="/A154948/b154948_1.txt">Rows n = 0..100 of triangle, flattened</a>
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#10 by Alois P. Heinz at Tue Feb 18 20:28:30 EST 2020
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Discussion
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| Tue Feb 18
| 20:28
| OEIS Server: Installed new b-file as b154948.txt. Old b-file is now b154948_1.txt.
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#9 by G. C. Greubel at Tue Feb 18 19:56:40 EST 2020
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#8 by G. C. Greubel at Tue Feb 18 19:55:28 EST 2020
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| LINKS
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G. C. Greubel, <a href="/A154948/b154948_1.txt">Rows n = 0..100 of triangle, flattened</a>
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| FORMULA
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T(n, k) = binomial(n+1,k+1)*2F1(-(n-k), -(k-1); -(n+1); -1). - G. C. Greubel, Feb 18 2020
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| EXAMPLE
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1,;
1, , 1,;
2, , 3, , 1,;
2, , 6, , 5, , 1,;
3, 10, 14, , 7, , 1,;
3, 15, 30, 26, , 9, , 1,;
4, 21, 55, 70, 42, 11, 1;
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| MAPLE
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seq(seq( add(binomial(k-1, j)*binomial(n-j+1, k+1), j=0..n+1), k=0..n), n=0..10); # G. C. Greubel, Feb 18 2020
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| MATHEMATICA
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Table[Binomial[n+1, k+1]*Hypergeometric2F1[-n+k, -k+1, -n-1, -1], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)
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| PROG
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(MAGMA) [ (&+[Binomial(k-1, j)*Binomial(n-j+1, k+1): j in [0..n+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020
(Sage) [[ sum(binomial(k-1, j)*binomial(n-j+1, k+1) for j in (0..n+1)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 18 2020
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| STATUS
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approved
editing
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#7 by Alois P. Heinz at Tue Feb 18 19:21:16 EST 2020
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#6 by Michel Marcus at Tue Feb 18 17:18:53 EST 2020
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#5 by Michel Marcus at Tue Feb 18 17:18:41 EST 2020
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| COMMENTS
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Row sums are A113300(n+1). Diagonal sums are A154949. Product of A154950 and A007318.
Product of A154950 and A007318.
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| FORMULA
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Number triangle T(n,k)=sum{) = Sum_{j=0..n+1, } C(n+1-j,k+1))*C(k-1,j)}.).
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| STATUS
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proposed
editing
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#4 by Georg Fischer at Tue Feb 18 16:05:46 EST 2020
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#3 by Georg Fischer at Tue Feb 18 16:05:06 EST 2020
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| DATA
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1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 3, 10, 14, 7, 1, 3, 15, 30, 26, 9, 1, 4, 21, 55, 70, 42, 11, 1, 4, 28, 91, 155, 138, 62, 13, 1, 5, 36, 140, 301, 363, 242, 86, 15, 1, 0, 5, 45, 204, 532, 819, 743, 390, 114, 17, 1, 6, 55, 285, 876, 1652, 1925, 1375, 590, 146, 19, 1
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| EXTENSIONS
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a(45)=0 removed by Georg Fischer, Feb 18 2020
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| STATUS
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approved
editing
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Discussion
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| Tue Feb 18
| 16:05
| Georg Fischer: Typo found by jOEIS testing.
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