A299427
Square table where T(n,k) = binomial(n*(n+k), k) * n/(n+k), for n>=1, k>=0, as read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 4, 1, 1, 9, 14, 1, 1, 16, 63, 48, 1, 1, 25, 184, 408, 165, 1, 1, 36, 425, 1872, 2565, 572, 1, 1, 49, 846, 6175, 17980, 15939, 2002, 1, 1, 64, 1519, 16536, 82775, 167552, 98670, 7072, 1, 1, 81, 2528, 38318, 292581, 1059380, 1535352, 610740, 25194, 1, 1, 100, 3969, 79808, 861175, 4874688, 13177125, 13934752, 3786588, 90440, 1
Offset: 0
This table begins:
n=1: [1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 4, 14, 48, 165, 572, 2002, ...];
n=3: [1, 9, 63, 408, 2565, 15939, 98670, ...];
n=4: [1, 16, 184, 1872, 17980, 167552, 1535352, ...];
n=5: [1, 25, 425, 6175, 82775, 1059380, 13177125, ...];
n=6: [1, 36, 846, 16536, 292581, 4874688, 78119454, ...];
n=7: [1, 49, 1519, 38318, 861175, 18008676, 358919022, ...];
n=8: [1, 64, 2528, 79808, 2214640, 56592320, 1367090208, ...];
n=9: [1, 81, 3969, 153117, 5132565, 157000275, 4507103601, ...];
...
Row generating functions R(x,n)^(n^2) begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
R(x,2)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + 572*x^5 + ...
R(x,3)^9 = 1 + 9*x + 63*x^2 + 408*x^3 + 2565*x^4 + 15939*x^5 + ...
R(x,4)^16 = 1 + 16*x + 184*x^2 + 1872*x^3 + 17980*x^4 + 167552*x^5 + ...
R(x,5)^25 = 1 + 25*x + 425*x^2 + 6175*x^3 + 82775*x^4 + 1059380*x^5 + ...
R(x,6)^36 = 1 + 36*x + 846*x^2 + 16536*x^3 + 292581*x^4 + 4874688*x^5 + ...
...
Related series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
where R(x,n)^m = Sum_{k>=0} C(m + n*k, k) * m/(m + n*k) * x^k.
...
-
{T(n,k) = binomial(n*(n+k), k) * n/(n+k) }
/* Print as a square table of first 9 rows */
for(n=1,9,print1("n="n": [",); for(k=0,8,print1(T(n,k),", ")); print1("...];");print(""))
/* Print as a Flattened table read by antidiagonals */
for(n=1,10,for(k=0,n,print1(T(n-k+1,k),", ")))
A299043
G.f. Sum_{n>=0} Series_Reversion( x*(1-x)^n )^n.
Original entry on oeis.org
1, 1, 2, 7, 33, 191, 1293, 9941, 85137, 801067, 8194281, 90367696, 1067146336, 13418399528, 178808377777, 2514944176091, 37204969293137, 577131827509491, 9362170099804501, 158438822236836110, 2791230865213193695, 51090157185364462103, 969892719975254406849, 19066076629590290124814, 387539455534509836620517, 8134022943287699194376826, 176073319016203896275830713
Offset: 0
G.f. A(x) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 191*x^5 + 1293*x^6 + 9941*x^7 + 85137*x^8 + 801067*x^9 + 8194281*x^10 + ...
such that
A(x) = 1 + x*R(x,2) + x^2*R(x,3)^4 + x^3*R(x,4)^9 + x^4*R(x,5)^16 + x^5*R(x,6)^25 + x^6*R(x,7)^36 + ...
where series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
and series R(x,n+1)^(n^2) begin:
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ...
R(x,3)^4 = 1 + 4*x + 18*x^2 + 88*x^3 + 455*x^4 + 2448*x^5 + ...
R(x,4)^9 = 1 + 9*x + 72*x^2 + 570*x^3 + 4554*x^4 + 36855*x^5 + ...
R(x,5)^16 = 1 + 16*x + 200*x^2 + 2320*x^3 + 26180*x^4 + 292448*x^5 + ...
R(x,6)^25 = 1 + 25*x + 450*x^2 + 7175*x^3 + 108100*x^4 + 1581255*x^5 + ...
R(x,7)^36 = 1 + 36*x + 882*x^2 + 18480*x^3 + 357399*x^4 + 6601644*x^5 + ...
...
-
{a(n) = my(A); A = sum(m=0,n+1, serreverse( x*(1-x)^m +x^2*O(x^n) )^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
-
{a(n) = if(n==0,1, sum(k=0,n, binomial(n*(n-k) + k,k) * (n-k)^2/(n*(n-k) + k) ) )}
for(n=0,30,print1(a(n),", "))
A299426
E.g.f. Sum_{n>=0} Series_Reversion( x*exp(-n*x) )^n.
Original entry on oeis.org
1, 1, 4, 39, 688, 18765, 722016, 37003267, 2426725120, 197569640889, 19498786969600, 2288864397602871, 314642887620065280, 50002239358837749061, 9086161251793519280128, 1870292375864728408312875, 432549223770637009573052416, 111598253780454986901562293489, 31918043775392401233962828169216, 10063176613379061071841096012386911, 3479934178969181147698311417364480000
Offset: 0
E.g.f. A(x) = 1 + x + 4*x^2/2! + 39*x^3/3! + 688*x^4/4! + 18765*x^5/5! + 722016*x^6/6! + 37003267*x^7/7! + 2426725120*x^8/8! + ...
such that
A(x) = 1 + x*W(x,1) + x^2*W(x,2)^4 + x^3*W(x,3)^9 + x^4*W(x,4)^16 + x^5*W(x,5)^25 + x^6*W(x,6)^36 + ...
where series W(x,n) = exp( x*W(x,n)^n ) begin:
W(x,1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + ...
W(x,2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + 14641*x^5/5! + ...
W(x,3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + 65536*x^5/5! + ...
W(x,4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + 194481*x^5/5! + ...
W(x,5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + 456976*x^5/5! + ...
W(x,6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 15625*x^4/4! + 923521*x^5/5! + ...
...
and series W(x,n)^n = Series_Reversion( x*exp(-n*x) ) / x begin:
W(x,1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + ...
W(x,2)^2 = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + ...
W(x,3)^3 = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + ...
W(x,4)^4 = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + ...
W(x,5)^5 = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + ...
W(x,6)^6 = 1 + 6*x + 108*x^2/2! + 3456*x^3/3! + 162000*x^4/4! + ...
...
and series W(x,n)^(n^2) begin:
W(x,1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + ...
W(x,2)^4 = 1 + 4*x + 32*x^2/2! + 400*x^3/3! + 6912*x^4/4! + ...
W(x,3)^9 = 1 + 9*x + 135*x^2/2! + 2916*x^3/3! + 83349*x^4/4! + ...
W(x,4)^16 = 1 + 16*x + 384*x^2/2! + 12544*x^3/3! + 524288*x^4/4! + ...
W(x,5)^25 = 1 + 25*x + 875*x^2/2! + 40000*x^3/3! + 2278125*x^4/4! + ...
W(x,6)^36 = 1 + 36*x + 1728*x^2/2! + 104976*x^3/3! + 7776000*x^4/4! + ...
...
-
{a(n) = my(A); A = sum(m=0,n+1, serreverse( x*exp(-m*x +x^2*O(x^n) ) )^m ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
-
{a(n) = if(n==0,1, sum(k=0,n, n!/k! * n^(k-1) * (n-k)^(k+1) ) )}
for(n=0,20,print1(a(n),", "))
Showing 1-3 of 3 results.