A101845
Triangle formed by left half of A101842, read by rows.
Original entry on oeis.org
1, 1, 3, 1, 7, 16, 1, 15, 61, 115, 1, 31, 206, 626, 1056, 1, 63, 659, 2989, 7554, 11774, 1, 127, 2052, 13308, 47349, 105099, 154624, 1, 255, 6297, 56935, 274677, 824331, 1660957, 2337507, 1, 511, 19162, 237862, 1518478, 5960818, 15747154, 29428654
Offset: 1
Triangle begins:
1,
1, 3,
1, 7, 16,
1, 15, 61, 115,
1, 31, 206, 626, 1056,
1, 63, 659, 2989, 7554, 11774,
...
-
A101842 := proc(n,k) option remember ; if k < -n or k >= n then 0 ; elif n = 1 then 1; else (n-k)*A101842(n-1,k-1)+A101842(n-1,k)+(n+k+1)*A101842(n-1,k+1) ; fi ; end: A101845 := proc(n,k) A101842(n,-n+k-1) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d, ",A101845(n,k)) ; od: od: # R. J. Mathar, Aug 07 2007
-
(* T is A101842 *)
T[n_, k_] /; -n <= k <= n-1 := T[n, k] = (n-k)*T[n-1, k-1]+T[n-1, k]+(n+k+1)* T[n-1, k+1];
T[1, -1] = T[1, 0] = 1; T[, ] = 0;
A101845[n_, k_] := T[n, k-n-1];
Table[A101845[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2024 *)
A102012
Triangle formed by right half of A101842, read by rows.
Original entry on oeis.org
1, 3, 1, 16, 7, 1, 115, 61, 15, 1, 1056, 626, 206, 31, 1, 11774, 7554, 2989, 659, 63, 1, 154624, 105099, 47349, 13308, 2052, 127, 1, 2337507, 1660957, 824331, 274677, 56935, 6297, 255, 1, 39984640, 29428654, 15747154, 5960818, 1518478, 237862
Offset: 1
Triangle begins:
1
3, 1
16, 7, 1
115, 61, 15, 1
1056, 626, 206, 31, 1
11774, 7554, 2989, 659, 63, 1
-
A101842 := proc(n,k) option remember ; if k < -n or k >= n then 0 ; elif n = 1 then 1; else (n-k)*A101842(n-1,k-1)+A101842(n-1,k)+(n+k+1)*A101842(n-1,k+1) ; fi ; end: A102012 := proc(n,k) A101842(n,k-1) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d, ",A102012(n,k)) ; od: od: # R. J. Mathar, Aug 07 2007
A373657
Triangle read by rows: Coefficients of the polynomials P(n, x) * EP(n, x), where P denote the signed Pascal polynomials and EP the Eulerian polynomials A173018.
Original entry on oeis.org
1, -1, 1, 1, -1, -1, 1, -1, -1, 8, -8, 1, 1, 1, 7, -27, 19, 19, -27, 7, 1, -1, -21, 54, 54, -276, 276, -54, -54, 21, 1, 1, 51, -25, -675, 1650, -1002, -1002, 1650, -675, -25, 51, 1, -1, -113, -372, 3436, -5125, -5013, 21216, -21216, 5013, 5125, -3436, 372, 113, 1
Offset: 0
Triangle starts:
[0] [ 1]
[1] [-1, 1]
[2] [ 1, -1, -1, 1]
[3] [-1, -1, 8, -8, 1, 1]
[4] [ 1, 7, -27, 19, 19, -27, 7, 1]
[5] [-1, -21, 54, 54, -276, 276, -54, -54, 21, 1]
[6] [ 1, 51, -25, -675, 1650, -1002, -1002, 1650, -675, -25, 51, 1]
-
PolyProd := proc(P, Q, len) local ep, eq, epq, CL, n, k;
ep := (n, x) -> simplify(add(Q(n, k)*x^k, k = 0..n)):
eq := (n, x) -> simplify(add(P(n, k)*x^k, k = 0..n)):
epq := (n, x) -> expand(ep(n, x) * eq(n, x)):
CL := p -> PolynomialTools:-CoefficientList(p, x);
seq(CL(epq(n, x)), n = 0..len); ListTools:-Flatten([%]) end:
PolyProd((n, k) -> (-1)^(n-k)*binomial(n, k), combinat:-eulerian1, 7);
Showing 1-3 of 3 results.