cp's OEIS Frontend

A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = m*T(n-1,k-1)+(k+1)*T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.

%I A350970 #54 Mar 12 2022 07:54:35
%S A350970 1,1,1,1,1,1,1,2,2,2,2,5,8,6,6,5,16,28,40,24,24,16,61,136,180,240,120,
%T A350970 120,61,272,662,1232,1320,1680,720,720,272,1385,3968,7266,12096,10920,
%U A350970 13440,5040,5040,1385,7936,24568,56320,83664,129024,100800,120960,40320,40320,7936,50521,176896,408360,814080,1023120,1491840,1028160,1209600,362880,362880
%N A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = m*T(n-1,k-1)+(k+1)*T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.
%C A350970 Triangle connects Euler numbers on left and factorial numbers on right.
%D A350970 A. Boutin, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), 252-254.
%D A350970 E. Estanave, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), pp. 117-118.
%H A350970 Alois P. Heinz, <a href="/A350970/b350970.txt">Rows n = 0..140, flattened</a>
%F A350970 If we ignore the n=0 row, then the e.g.f. for column 0 is sec(x)+tan(x), and for column k >= 1 it is sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)). See the initial rows of the square array in the EXAMPLES section. - _N. J. A. Sloane_, Mar 05 2022
%F A350970 abs(Sum_{k=0..n} (-1)^k * T(n,k)) = A007836(n) for n>=2. - _Alois P. Heinz_, Mar 04 2022
%e A350970 Triangle begins:
%e A350970     1,
%e A350970     1,    1,
%e A350970     1,    1,    1,
%e A350970     1,    2,    2,    2,
%e A350970     2,    5,    8,    6,     6,
%e A350970     5,   16,   28,   40,    24,    24,
%e A350970    16,   61,  136,  180,   240,   120,   120,
%e A350970    61,  272,  662, 1232,  1320,  1680,   720,  720,
%e A350970   272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040,
%e A350970   ...
%e A350970 This may also be constructed as a square array, with entries T(n,k), n >= 1, 0 <= k, whose columns have e.g.f. equal to sec(x)+tan(x) (if k=0) and sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)) (if k>0):
%e A350970 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A350970 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A350970 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A350970 2, 5, 8, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A350970 5, 16, 28, 40, 24, 24, 0, 0, 0, 0, 0, 0, 0, ...
%e A350970 16, 61, 136, 180, 240, 120, 120, 0, 0, 0, 0, 0, 0, ...
%e A350970 61, 272, 662, 1232, 1320, 1680, 720, 720, 0, 0, 0, 0, 0, ...
%e A350970 272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040, 0, 0, 0, 0, ...
%e A350970 ...
%p A350970 for n from 0 to 12 do
%p A350970 T[n]:=Array(0..n,0);
%p A350970 T[0,0] := 1;
%p A350970 T[1,0] := 1; T[1,1] := 1;
%p A350970 if n>1 then
%p A350970   T[n,0] := T[n-1,1];
%p A350970 for k from 1 to n-2 do
%p A350970 m:=k; if ((n+k) mod 2) = 0 then m:=k-1; fi;
%p A350970 T[n,k] := m*T[n-1,k-1] + (k+1)*T[n-1,k+1];
%p A350970 od:
%p A350970 T[n,n-1] := (n-1)*T[n-1,n-2];
%p A350970 T[n,n] := T[n,n-1];
%p A350970 fi;
%p A350970 lprint( [seq(T[n,k],k=0..n)] );
%p A350970 od:
%p A350970 # second Maple program:
%p A350970 b:= proc(n, i) option remember; `if`(i=0,
%p A350970      `if`(n=0, 1, 0), b(n, i-1)+b(n-1, n-i))
%p A350970     end:
%p A350970 T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
%p A350970      `if`(k=0, b(n-1$2), `if`(n-k<=1, (n-1)!, (k+1)*
%p A350970       T(n-1, k+1)+(k-irem(1+n+k, 2))*T(n-1, k-1))))
%p A350970     end:
%p A350970 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Mar 04 2022
%p A350970 # To produce the square array, _N. J. A. Sloane_, Mar 05 2022:
%p A350970 read(transforms):
%p A350970 myegf := (f,M) -> SERIESTOLISTMULT(series(f,x,M));
%p A350970 T:=proc(n,k,M) local i;
%p A350970 if k=0 then myegf((sec(x)+tan(x)),M)[n];
%p A350970 else
%p A350970 myegf(sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)),M)[n];
%p A350970 fi;
%p A350970 end;
%p A350970 [seq(T(n,0,16),n=1..5)];
%p A350970 for n from 1 to 8 do
%p A350970 lprint([seq(T(n,k,16),k=0..12)]);
%p A350970 od:
%t A350970 b[n_, i_] := b[n, i] = If[i == 0,
%t A350970      If[n == 0, 1, 0], b[n, i - 1] + b[n - 1, n - i]];
%t A350970 T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
%t A350970      If[k == 0, b[n - 1, n - 1], If[n - k <= 1, (n - 1)!, (k + 1)*
%t A350970      T[n - 1, k + 1] + (k - Mod[1 + n + k, 2])*T[n - 1, k - 1]]]];
%t A350970 Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 12 2022, _Alois P. Heinz_ *)
%Y A350970 The initial columns are A000111, A000111, A225689, A350971.
%Y A350970 The diagonals, reading from the right, are (essentially) A000142, A000142, A002301, A006157, A002302, A350973, A002303, A350974, A350975.
%Y A350970 Rows sums give A156142(n-1).
%Y A350970 Cf. A007836.
%K A350970 nonn,tabl
%O A350970 0,8
%A A350970 _N. J. A. Sloane_, Mar 03 2022