A193194
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (1+x)^(n*(n+1)/2), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.
Original entry on oeis.org
1, 1, 1, 3, 20, 262, 5367, 158413, 6318384, 326575077, 21199737973, 1687053244236, 161418184139781, 18276066372054109, 2416167457088427950, 368773198369779785338, 64348161941454274119082, 12728047101293068225626576, 2832615019902477894227329544
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 20*x^5 + 262*x^6 + 5367*x^7 +...
where
A(A(x)) = x*(1+x) + x^2*(1+x)^3 + x^3*(1+x)^6 + 3*x^4*(1+x)^10 + 20*x^5*(1+x)^15 + 262*x^6*(1+x)^21 +...+ a(n)*x^n*(1+x)^(n*(n+1)/2) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 4*x^3 + 12*x^4 + 66*x^5 + 717*x^6 + 13344*x^7 +...
-
{a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);
G=sum(m=1,#A-1,A[m]*x^m*(1+x+x*O(x^#A))^(m*(m+1)/2));
A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}
A193192
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n * (1+x)^(n^2), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.
Original entry on oeis.org
1, 1, 2, 13, 184, 4725, 188596, 10765407, 829780846, 82924007284, 10420182259194, 1607406366386249, 298555458341808338, 65711158773953092780, 16910051487116790543954, 5030141451818448773854244, 1712632076858599057432066794
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 13*x^4 + 184*x^5 + 4725*x^6 +...
where
A(A(x)) = x*(1+x) + x^2*(1+x)^4 + 2*x^3*(1+x)^9 + 13*x^4*(1+x)^16 + 184*x^5*(1+x)^25 + 4725*x^6*(1+x)^36 +...+ a(n)*x^n*(1+x)^(n^2) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 6*x^3 + 37*x^4 + 468*x^5 + 11054*x^6 + 421428*x^7 +...
-
{a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);
G=sum(m=1,#A-1,A[m]*x^m*(1+x+x*O(x^#A))^(m^2));
A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}
A193195
G.f. satisfies: A(A(x)) = Sum_{n>=1} a(n)*x^n / (1-x)^(n*(n+1)/2), where g.f. A(x) = Sum_{n>=1} a(n)*x^n.
Original entry on oeis.org
1, 1, 2, 8, 63, 866, 18444, 559083, 22773527, 1197061138, 78782852673, 6341384941543, 612605031308910, 69931195961966196, 9310803519433216321, 1429869869684956113511, 250857267705012344767575, 49858270430813771746874366, 11143529422156562195864991584
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 63*x^5 + 866*x^6 + 18444*x^7 +...
where
A(A(x)) = x/(1-x) + x^2/(1-x)^3 + 2*x^3/(1-x)^6 + 8*x^4/(1-x)^10 + 63*x^5/(1-x)^15 + 866*x^6/(1-x)^21 +...+ a(n)*x^n/(1-x)^(n*(n+1)/2) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 196*x^5 + 2379*x^6 + 46224*x^7 +...
-
{a(n)=local(A=[1],F=x,G=x);for(i=1,n,A=concat(A,0);F=x*Ser(A);
G=sum(m=1,#A-1,A[m]*x^m/(1-x+x*O(x^#A))^(m*(m+1)/2));
A[#A]=Vec(G)[#A]-Vec(subst(F,x,F))[#A]);if(n<1,0,A[n])}
A182042
Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, 0, -3/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Original entry on oeis.org
1, 0, 3, 0, 6, 9, 0, 9, 27, 27, 0, 12, 54, 108, 81, 0, 15, 90, 270, 405, 243, 0, 18, 135, 540, 1215, 1458, 729, 0, 21, 189, 945, 2835, 5103, 5103, 2187, 0, 24, 252, 1512, 5670, 13608, 20412, 17496, 6561, 0, 27, 324, 2268, 10206, 30618, 61236, 78732, 59049, 19683
Offset: 0
Triangle begins:
1;
0, 3;
0, 6, 9;
0, 9, 27, 27;
0, 12, 54, 108, 81;
0, 15, 90, 270, 405, 243;
0, 18, 135, 540, 1215, 1458, 729;
0, 21, 189, 945, 2835, 5103, 5103, 2187;
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T:= proc(n, k) option remember;
if k=n then 3^n
elif k=0 then 0
else binomial(n,k)*3^k
fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020
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With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x+x^2+3*y*x^2)/(1-2*x-3*y*x+x^2+3*y*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)
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T(n,k) = if (k==0, 1, binomial(n,k)*3^k);
matrix(10, 10, n, k, T(n-1,k-1)) \\ to see the triangle \\ Michel Marcus, Feb 17 2020
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@CachedFunction
def T(n, k):
if (k==n): return 3^n
elif (k==0): return 0
else: return binomial(n,k)*3^k
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 17 2020
Showing 1-4 of 4 results.
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