A105557
Row sums of triangle A105556, in which column n equals the row sums of A001263^n, which is the n-th matrix power of the Narayana triangle A001263.
Original entry on oeis.org
1, 2, 4, 10, 32, 128, 626, 3681, 25574, 206402, 1908996, 20024149, 236142157, 3106393358, 45265833590, 726249472784, 12761749378320, 244453274012442, 5082582988294164, 114258645210526486, 2767462674168199303
Offset: 0
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{a(n)=local(N=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1,j-1)*binomial(m,j-1)/j))); sum(k=0,n,sum(j=0,n-k,(N^k)[n-k+1,j+1]))}
A105558
Central terms in even-indexed rows of triangle A105556 and thus equals the n-th row sum of the n-th matrix power of the Narayana triangle A001263.
Original entry on oeis.org
1, 2, 12, 148, 3105, 99156, 4481449, 272312216, 21414443481, 2116193061340, 256712977920256, 37506637787774112, 6496315164318118165, 1316230822119433518312, 308426950979497974254310
Offset: 0
Terms a(n) divided by (n+1) begin:
1,1,4,37,621,16526,640207,34039027,2379382609,211619306134,...
Contribution from _Paul D. Hanna_, Jan 31 2009: (Start)
G.f.: A(x) = 1 + 2*x + 12*x^2/3 + 148*x^3/18 + 3105*x^4/180 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and:
F(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ A155926(n)*x^n/[n!*(n+1)!/2^n] +...
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 +...+ x^n/[n!*(n+1)!/2^n] +... (End)
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a(n)=local(N=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1,j-1)*binomial(m,j-1)/j))); sum(j=0,n,(N^n)[n+1,j+1])
for(n=0,20,print1(a(n),", "))
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a(n)=local(F=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(deriv(serreverse(x/F)),n)*n!*(n+1)!/2^n
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Jan 31 2009
A155926
G.f. satisfies: A(x) = B(x*A(x)) where A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n].
Original entry on oeis.org
1, 1, 4, 37, 621, 16526, 640207, 34039027, 2379382609, 211619306134, 23337543447296, 3125553148981176, 499716551101393705, 94016487294245251308, 20561796731966531616954, 5172827581575899147920471
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
B(x) = 1 + x + 1/3*x^2 + 1/18*x^3 + 1/180*x^4 +...+ x^n/[n!*(n+1)!/2^n] +... where
A(x) = B(x*A(x)) and B(x) = A(x/B(x)) ;
1/B(x) = 1 - x + 2*x^2/3 - 7*x^3/18 + 39*x^4/180 - 321*x^5/2700 +...+ (-1)^n*A103365(n)*x^n/[n!*(n+1)!/2^n] +...
Also, A(x) = C(x*A(x)^2) where:
C(x) = 1 + x - 2*x^2/3 + 19*x^3/18 - 379*x^4/180 + 12726*x^5/2700 +...+ A155927(n)*x^(n+1)/[n!*(n+1)!/2^n] +...
A(x)^2 = 1 + 2*x + 11*x^2/3 + 122*x^3/18 + 2302*x^4/180 + 66482*x^5/2700 +...
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{a(n)=local(F=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(serreverse(x/F)/x,n)*n!*(n+1)!/2^n}
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{a(n)=local(N=matrix(n+1, n+1, m, j, if(m>=j, binomial(m-1, j-1)*binomial(m, j-1)/j))); sum(j=0, n, (N^n)[n+1, j+1])/(n+1)}
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