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.
Table starts: 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 3 7 17 41 99 239 577 1393 3363 8119 19601 47321 114243 275807 4 10 26 68 178 466 1220 3194 8362 21892 57314 150050 392836 1028458 5 13 35 95 259 707 1931 5275 14411 39371 107563 293867 802859 2193451 6 16 44 122 340 950 2658 7442 20844 58392 163594 458356 1284250 3598338 7 19 53 149 421 1193 3387 9627 27383 77923 221805 631469 1797957 5119593 8 22 62 176 502 1436 4116 11814 33942 97582 280676 807574 2324116 6689624 9 25 71 203 583 1679 4845 14001 40503 117263 339699 984515 2854281 8277153 10 28 80 230 664 1922 5574 16188 47064 136946 398746 1161634 3385486 9869934 11 31 89 257 745 2165 6303 18375 53625 156629 457795 1338779 3916897 11463989 Some solutions for 5 X 3: 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
rows = 11; rowGf[n_, x_] = 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 + ChebyshevU[n-1, (1-x)/(2*x)])/ChebyshevU[n, (1-x)/(2*x)])/(1-3*x)^2;
row[n_] := rowGf[n+1, x] + O[x]^(rows+1) // CoefficientList[#, x]& // Rest; T = Array[row, rows]; Table[T[[n-k+1, k]], {n, 1, rows}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)
\\ from Knopfmacher et al.
RowGf(k, x='x) = my(z=(1-x)/(2*x)); 1 + (x*(k-(3*k+2)*x) + (2*x^2)*(1+polchebyshev(k-1, 2, z))/polchebyshev(k, 2, z))/(1-3*x)^2;
T(n,k) = {polcoef(RowGf(n+1) + O(x*x^k),k)}
for(n=1, 10, print(Vec(RowGf(n+1) + O(x^11)))) \\ Andrew Howroyd, Apr 15 2017 [updated Mar 13 2021]
All solutions for n=3: ..1....1....6....3....6....5....2....4....4....3....5....4....5....7....6....2 ..2....1....6....4....7....5....3....4....5....3....6....4....5....7....6....2 ..2....1....7....4....7....5....3....5....5....3....6....4....6....7....6....3 .. ..1....2....3 ..1....2....3 ..2....2....4
Triangle begins: 1; 2, 4; 3, 7, 17; 4, 10, 26, 68; 5, 13, 35, 95, 259; 6, 16, 44, 122, 340, 950; 7, 19, 53, 149, 421, 1193, 3387; 8, 22, 62, 176, 502, 1436, 4116, 11814; 9, 25, 71, 203, 583, 1679, 4845, 14001, 40503; 10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946;
Inm := proc(n,m)
if m >= n then
(n+2)*3^(n-2)+(m-n)*add(A005773(i)*A005773(n-i),i=0..n-1)
+2*add((n-k-2)*3^(n-k-3)*A001006(k),k=0..n-3) ;
else
0 ;
end if;
end proc:
for m from 1 to 13 do
for n from 1 to m do
printf("%a,",Inm(n,m)) ;
end do:
printf("\n") ;
end do:
# Second program:
A296449row := proc(n) local gf, ser;
gf := n -> 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 +
ChebyshevU(n - 1, (1 - x)/(2*x))) / ChebyshevU(n, (1 - x)/(2*x)))/(1 - 3*x)^2;
ser := n -> series(expand(gf(n)), x, n + 1);
seq(coeff(ser(n), x, k), k = 1..n) end:
for n from 0 to 11 do A296449row(n) od; # Peter Luschny, Sep 07 2021
(* b = A005773 *) b[0] = 1; b[n_] := Sum[k/n*Sum[Binomial[n, j] * Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}]; (* c = A001006 *) c[0] = 1; c[n_] := c[n] = c[n-1] + Sum[c[k] * c[n-2-k], {k, 0, n-2}]; Inm[n_, m_] := If[m >= n, (n + 2)*3^(n - 2) + (m - n)*Sum[b[i]*b[n - i], {i, 0, n - 1}] + 2*Sum[(n - k - 2)*3^(n - k - 3)*c[k], {k, 0, n-3}], 0]; Table[Inm[n, m], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 23 2018, adapted from first Maple program. *)
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