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.
Gottfried Helms has authored 48 sequences. Here are the ten most recent ones:
f(x) = x + 1/4*x^2 + 1/48*x^3 + 1/3840*x^5 - 7/92160*x^6 + 1/645120*x^7 + O(x^8) so c_3 = 1/48 and a(3) = c_3 * 4^2*3! = 16*6/48 = 2
max = 19; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; coes = CoefficientList[ Series[f[f[x]] - Exp[x] - 1, {x, 0, max}], x]; sol = Solve[Thread[coes == 0] // Rest] // First; Table[c[n]*4^(n-1)*n!, {n, 0, max}] /. sol (* Jean-François Alcover, Feb 11 2013 *)
{a(n)=local(A=x+x^2,B=x);for(i=1,n,B=serreverse(A+x*O(x^n));A=(A+exp(B)-1)/2);4^(n-1)*n!*polcoeff(A,n)} \\ Paul D. Hanna
{trisqrt(m) = local(tmp, rs=rows(m), cs=cols(m), c);
\\ computes sqrt of lower triangular matrix with unit-diagonal
tmp=matid(#m);
for(d=1,rs-1,
for(r=d+1,rs,
c=r-d;
tmp[r,c]=(m[r,c]-sum(k=c+1,r-1,tmp[r,k]*tmp[k,c]))
/(tmp[c,c]+tmp[r,r])
);
);
return(tmp);}
ff = exp(x)-1
Mff = matrix(6,6,r,c,polcoeff(ff^(c-1),(r-1))) \\ create Bell-matrix for ff
Mf = trisqrt ( Mff ) \\ = Mff^(1/2) is Bellmatrix for f
f = Ser(Mf[,2]) \\ coefficients of power series for half-iterate of exp(x)-1 from second column in Mf
Triangle begins: 1 1 1 1 1 1 3 4 6 4 3 1 1 7 13 26 31 31 25 13 6 1 1 15 40 100 171 220 255 215 156 85 35 10 1
Triangle begins: ..........................1 .....................1....1....1 ................1....3....4....3....1 ...........1....7...13...19...13....6...1 ......1...15...40...85...96...75...35..10..1 ..1..31..121..335..560..616..471..240..80..15..1 ................................................. Assume a matrix-function rowshift(M) which computes M1 = rowshift(M) in the following way: M = [a,b,c,...] [k,l,m,...] [r,s,t,...] [.........] becomes M1 = [a,b,c, ......] [0,k,l,m, ....] [0,0,r,s,t,...] [ ............] Define the lower-triangular matrix of Stirling-numbers of the second kind S = [1 0 0 0 ...] [1 1 0 0 ...] [1 3 1 0 ...] [1 7 6 1 ...] [ ..........] Then with H0 = [1] [1] [1] [1] ... we have H1 = S * rowshift(H0) \\ = S H2 = S * rowshift(H1) H3 = S * rowshift(H2) ... H1 = 1 . . . . 1 1 . . . 1 3 1 . . 1 7 6 1 . 1 15 25 10 1 H2= 1 . . . . . . . . 1 1 1 . . . . . . 1 3 4 3 1 . . . . 1 7 13 19 13 6 1 . . 1 15 40 85 96 75 35 10 1 H3= 1 . . . . . . . . . . . . 1 1 1 1 . . . . . . . . . 1 3 4 6 4 3 1 . . . . . . 1 7 13 26 31 31 25 13 6 1 . . . 1 15 40 100 171 220 255 215 156 85 35 10 1 (based on the Maple implementation from _R. J. Mathar_)
# From R. J. Mathar: (Start) X := proc(k,l,n) if k >=1 and k <=n and l >=1 and l <= n then combinat[stirling2](n,k)*combinat[stirling2](k,l) ; else 0 ; fi ; end: H := proc(n,j) add( X(j-l,l,n),l=1..floor(j/2)) ; end: for n from 1 to 10 do for j from 2 to 2*n do printf("%d ",H(n,j)) ; od: printf("\n") ; od: # (End)
A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A129323 := proc(n) A078937(n+1,1) ; end: seq(A129323(n),n=0..23) ; # R. J. Mathar, May 30 2008
Table[Sum[BellB[n, 2], {i, 0, n}], {n, -1, 20}] (* Zerinvary Lajos, Jul 16 2009 *)
A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129327 := proc(n) A078938(n+1,1) ; end: seq(A129327(n),n=0..27) ; # R. J. Mathar, May 30 2008
Table[Sum[BellB[n, 3], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)
A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129331 := proc(n) A078939(n+1,1) ; end: seq(A129331(n),n=0..25) ; # R. J. Mathar, May 30 2008
Table[Sum[BellB[n, 4], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)
A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A129325 := proc(n) A078937(n+1,3) ; end: seq(A129325(n),n=0..27) ; # R. J. Mathar, May 30 2008
A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]]; A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}]; a[n_] := A078937[n + 1, 3]; a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)
m=matpascal(30)-matid(31); pe=matid(31)+sum(i=1,30,m^i/i!); A=pe^2; A[,4] \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129328 := proc(n) A078938(n+1,2) ; end: seq(A129328(n),n=0..27) ; # R. J. Mathar, May 30 2008
A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]]; A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}]; A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}]; a[n_] := A078938[n + 1, 2]; a /@ Range[0, 20] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)
A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129332 := proc(n) A078939(n+1,2) ; end: seq(A129332(n),n=0..25) ; # R. J. Mathar, May 30 2008
A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]]; A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}]; A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}]; A078939[n_, c_] := Sum[A078938[n, k] A056857[k + 1, c], {k, 0, n}]; a[n_] := A078939[n + 1, 2]; a /@ Range[0, 19] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)
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