Revision History for A104980
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Showing entries 1-10
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| A104980
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Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T), or [T^p](m,0) = p*T(p+m,p+1) for all m>=1 and p>=-1.
(history;
published version)
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#23 by Alois P. Heinz at Mon Jun 07 15:45:29 EDT 2021
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#22 by G. C. Greubel at Mon Jun 07 15:44:29 EDT 2021
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#21 by G. C. Greubel at Mon Jun 07 15:43:53 EDT 2021
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| DATA
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1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 21, 5, 1, 3447, 1297, 461, 133, 31, 6, 1, 29093, 10063, 3447, 977, 225, 43, 7, 1, 273343, 87669, 29093, 8135, 1859, 353, 57, 8, 1, 2829325, 847015, 273343, 75609, 17185, 3251, 523, 73, 9, 1, 31998903
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| LINKS
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G. C. Greubel, <a href="/A104980/b104980.txt">Rows n = 0..50 of the triangle, flattened</a>
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| EXAMPLE
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1;
1,, 1;
3,, 2,, 1;
13,, 7,, 3,, 1;
71,, 33,, 13,, 4,, 1;
461,, 191,, 71,, 21,, 5,, 1;
3447,, 1297,, 461,, 133,, 31,, 6,, 1;
29093,, 10063,, 3447,, 977,, 225,, 43,, 7,, 1;
273343,, 87669,, 29093,, 8135,, 1859,, 353,, 57,, 8,, 1;
2829325,, 847015,, 273343,, 75609,, 17185,, 3251,, 523,, 73,, 9,, 1; ...
1;
- -1,, 1;
- -1,-, -2,, 1;
- -3,-, -1,-, -3,, 1;
- -13,-, -3,-, -1,-, -4,, 1;
- -71,-, -13,-, -3,-, -1,-, -5,, 1;
- -461,-, -71,-, -13,-, -3,-, -1,-, -6,, 1; ...
1;
1,, 1;
2,, 1,, 1;
7,, 3,, 1,, 1;
33,, 13,, 4,, 1,, 1;
191,, 71,, 21,, 5,, 1,, 1; ...
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| MATHEMATICA
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T[n_, k_] := _]:= T[n, k] = ]= If[n<k || k<0, 0, If[n == ==k, 1, If[n == ==k+1, n, k T[n, k+1] + Sum[T[j, 0] T[n, j+k+1], {j, 0, n-k-1}]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // }]//Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)
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| PROG
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(PARI) {T(n, k) = if(n<k||k<0, , 0, , if(n==k, , 1, , if(n==k+1, , n, k*T(n, k+1)+) + sum(j=0, n-k-1, T(j, 0)*T(n, j+k+1)))))}
for(n=0, , 10, , for(k=0, , n, , print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = if(n<k||k<0, , 0, (, (matrix(n+1, , n+1, , m, , j, , if(m==j, , 1, , if(m==j+1, -, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x+O(x^m), m-j-1))))^-1)[n+1, k+1])}
(Sage)
@CachedFunction
def T(n, k):
if (k<0 or k>n): return 0
elif (k==n): return 1
elif (k==n-1): return n
else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 07 2021
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| STATUS
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approved
editing
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#20 by Bruno Berselli at Thu Aug 09 02:46:55 EDT 2018
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#19 by Michel Marcus at Thu Aug 09 02:26:26 EDT 2018
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#18 by Michel Marcus at Thu Aug 09 02:26:21 EDT 2018
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| COMMENTS
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Contribution fromFrom Paul D. Hanna, Feb 17 2009: (Start)
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| STATUS
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proposed
editing
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#17 by Jean-François Alcover at Thu Aug 09 02:05:05 EDT 2018
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#16 by Jean-François Alcover at Thu Aug 09 02:05:00 EDT 2018
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| MATHEMATICA
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T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[n == k, 1, If[n == k+1, n, k T[n, k+1] + Sum[T[j, 0] T[n, j+k+1], {j, 0, n-k-1}]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)
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approved
editing
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#15 by N. J. A. Sloane at Thu Jul 12 00:34:42 EDT 2018
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#14 by Michel Marcus at Thu Jul 12 00:26:09 EDT 2018
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