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A331432
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Triangle T(n,k) (n >= k >= 0) read by rows: T(n,0) = (1+(-1)^n)/2; for k>=1, set T(0,k) = 0, S(n,k) = binomial(n,k)*binomial(n+k+1,k), and for n>=1, T(n,k) = S(n,k)-T(n-1,k).
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6
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1, 0, 3, 1, 5, 10, 0, 10, 35, 35, 1, 14, 91, 189, 126, 0, 21, 189, 651, 924, 462, 1, 27, 351, 1749, 4026, 4290, 1716, 0, 36, 594, 4026, 13299, 22737, 19305, 6435, 1, 44, 946, 8294, 36751, 89375, 120835, 85085, 24310, 0, 55, 1430, 15730, 89375, 289003, 551837, 615043, 369512, 92378, 1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716
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OFFSET
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0,3
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COMMENTS
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The scanned pages of Ser are essentially illegible, and the book is out of print and hard to locate.
For Table IV on page 93, it is simplest to ignore the minus signs. The present triangle then matches all the given terms in that triangle, so it seems best to define the triangle by the recurrences given here, and to conjecture (strongly) that this is the same as Ser's triangle.
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REFERENCES
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J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
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LINKS
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FORMULA
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T(n, k) = binomial(n,k)*binomial(n+k+1,k) - T(n-1, k), with T(n, 0) = (1 + (-1)^n)/2.
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EXAMPLE
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Triangle begins:
1;
0, 3;
1, 5, 10;
0, 10, 35, 35;
1, 14, 91, 189, 126;
0, 21, 189, 651, 924, 462;
1, 27, 351, 1749, 4026, 4290, 1716;
0, 36, 594, 4026, 13299, 22737, 19305, 6435;
1, 44, 946, 8294, 36751, 89375, 120835, 85085, 24310;
0, 55, 1430, 15730, 89375, 289003, 551837, 615043, 369512, 92378;
1, 65, 2080, 27950, 197275, 811733, 2047123, 3203837, 3031678, 1587222, 352716;
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MAPLE
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SS := (n, k)->binomial(n, k)*binomial(n+k+1, k);
T4:=proc(n, k) local i; global SS; option remember;
if k=0 then return((1+(-1)^n)/2); fi;
if n=0 then 0 else SS(n, k)-T4(n-1, k); fi; end;
rho:=n->[seq(T4(n, k), k=0..n)];
for n from 0 to 14 do lprint(rho(n)); od:
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MATHEMATICA
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T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0, (1 + (-1)^n)/2, Binomial[n, k]*Binomial[n+k+1, k] - T[n-1, k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)
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PROG
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(Sage)
if (n<0): return 0
elif (k==0): return ((n+1)%2)
else: return binomial(n, k)*binomial(n+k+1, k) - T(n-1, k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022
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CROSSREFS
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Taking the component-wise sums of the rows by pairs give the triangle in A178303.
Ser's tables I and III are A331430 and A331431 (both are still mysterious).
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KEYWORD
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AUTHOR
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STATUS
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approved
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