This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
A(x;t) = 1 + 2*x + (5 + 2*t)*x^2 + (14 + 14*t + 2*t^2)*x^3 + ... Triangle starts: n\k | 0 1 2 3 4 5 6 7 8 -----+----------------------------------------------------------- 0 | 1; 1 | 2; 2 | 5, 2; 3 | 14, 14, 2; 4 | 42, 72, 27, 2; 5 | 132, 330, 220, 44, 2; 6 | 429, 1430, 1430, 520, 65, 2; 7 | 1430, 6006, 8190, 4550, 1050, 90, 2; 8 | 4862, 24752, 43316, 33320, 11900, 1904, 119, 2; 9 | 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2;
T(n,k):=(binomial(n-1,k)*binomial(2*(n+1),n-k))/(n+1); /* Vladimir Kruchinin, Jan 14 2022 */
A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x; A286785_ser(N,t='t) = 1/(1-x*A286784_ser(N,t))^2; concat(apply(p->Vecrev(p), Vec(A286785_ser(12))))
V := proc(r,k)
local a ,t;
a := k-1;
for t from k-2 to 1 by -1 do
a := a*(r+t)/(t+2) ;
end do:
for t from 3 to k+1 do
a := a*(r-t)/(k-t+2) ;
end do:
a ;
end proc:
A002060 := proc(n)
V(n,n-5) ;
end proc:
seq(A002060(n),n=7..25) ; # R. J. Mathar, Nov 26 2011
V[r_, k_] := Module[{a, t}, a = k - 1;
For[t = k - 2, t >= 1, t--, a = a*(r + t)/(t + 2)];
For[t = 3, t <= k + 1, t++, a = a*(r - t)/(k - t + 2)];
a];
A002060[n_] := V[n, n - 5];
Table[A002060[n], {n, 7, 29}] (* Jean-François Alcover, Mar 10 2023, after R. J. Mathar *)
Triangle begins:
1;
2, 4;
3, 14, 14;
4, 32, 72, 48;
5, 60, 225, 330, 165;
6, 100, 550, 1320, 1430, 572;
...
V := proc(n,x)
local X,g,i ;
X := x^2/(1-x) ;
g := X^n ;
for i from 1 to n-2 do
g := diff(g,x) ;
end do;
x^2*g*2*(n-1)/n! ;
end proc;
A259476 := proc(n,k)
V(k-n+2,x) ;
coeftayl(%,x=0,n+2) ;
end proc:
for n from 4 to 14 do
for k from n to 2*n-4 do
printf("%d,",A259476(n,k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, Jul 09 2015
T[n_, m_] := 2 Binomial[m, n] Binomial[n-2, m-n+2]/(n-2);
Table[T[n, m], {n, 4, 14}, {m, n, 2n-4}] // Flatten (* Jean-François Alcover, Apr 15 2023, after Vladimir Kruchinin *)
T(n,m):=if n<4 then 0 else (2*binomial(m,n)*binomial(n-2,m-n+2))/(n-2); /* Vladimir Kruchinin, Jan 27 2022 */
1; 1,4; 1,12,6; 1,40,20,8; 1,140,70,30,10; 1,504,252,112,42,12; 1,1848,924,420,168,56,14; 1,6864,3432,1584,660,240,72,16; 1,25740,12870,6006,2574,990,330,90,18; 1,97240,48620,22880,10010,4004,1430,440,110,20;
A217234_row := proc(n) local A,r,c,s ; A := array({},1..n,1..n) ; for r from 2 to n do A[r,1] := A[1,r] ; end do: for r from 2 to n do for c from 2 to n do A[r,c] := A[r,c-1]+A[r-1,c] ; end do: end do: s := add(add( A[r,c],c=1..n) ,r=1..n) ; for c from 1 to n do printf("%d,", coeff(s,A[1,c]) ) ; end do: return ; end proc: for n from 1 to 10 do A217234_row(n) ; printf(";\n") ; end do; # R. J. Mathar, Oct 13 2012
A217234row [n_] := Module[{A, x, r, c, s }, A = Array[x, {n, n}]; Do[A[[r, 1]] = A[[1, r]], {r, 2, n}]; Do[A[[r, c]] = A[[r, c - 1]] + A[[r - 1, c]], {r, 2, n}, {c, 2, n}]; s = Sum[A[[r, c]], {r, 1, n}, {c, 1, n}]; If[n == 1, {1}, List @@ s /. x[, ] -> 1]]; Table[A217234row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 04 2023, after R. J. Mathar *)
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