A120021 -id:A120021 - cp's OEIS frontend

A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4x))/2 o x/(1-nx) o (x-x^2) } for n>=1.

1, 2, 9, 68, 710, 9348, 148085, 2740672, 58033953, 1383923040, 36705564368, 1071911496576, 34179790156473, 1181725089179936, 44035415728886145, 1759481180119564288, 75042973200676887772, 3402984761691650083008

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

Main diagonal of A120019, the table of self-compositions of A120010.

For n>=1, n divides a(n): a(n)/n = A120022(n).

			Successive iterations of F(x), the g.f. of A120010, begin:
F(x) = (1)x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = x + (2)x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + 3x^2 + (9)x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + (68)x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + (710)x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + (9348)x^6+..

Cf. A120010, A120019, A120021, A120022.

{a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-1)))}
for(n=1,25,print1(a(n),", "))

a(n) = Sum_{j=1..n} Catalan(n-j) * [ Sum_{i=1..j} (-1)^(j-i) * n^(i-1) * C(n-j+i, j-i) * C(n-j+i-1, i-1) ];

a(n) = Sum_{j=0..n-1} n^j * [ Sum_{i=j..n-1} (-1)^(i-j) * Catalan(n-i-1) * C(n-i+j, i-j) * C(n-i+j-1, j) ], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

A120019 Square table, read by antidiagonals, of self-compositions of A120010.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 9, 10, 6, 1, 5, 16, 30, 32, 18, 1, 6, 25, 68, 114, 116, 53, 1, 7, 36, 130, 312, 480, 440, 158, 1, 8, 49, 222, 710, 1536, 2157, 1708, 481, 1, 9, 64, 350, 1416, 4070, 8000, 10092, 6760, 1491, 1, 10, 81, 520, 2562, 9348, 24365, 43472, 48525

Offset: 1

Paul D. Hanna, Jun 14 2006

The g.f. of row n is the composition: (1-sqrt(1-4*x))/2 o x/(1-nx) o (x-x^2).

			Square table begins:
1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, ...
1, 2, 4, 10, 32, 116, 440, 1708, 6760, 27232, ...
1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, ...
1, 4, 16, 68, 312, 1536, 8000, 43472, 243808, 1400448, ...
1, 5, 25, 130, 710, 4070, 24365, 151330, 968785, 6355795, ...
1, 6, 36, 222, 1416, 9348, 63768, 448188, 3234216, 23875296, ...
1, 7, 49, 350, 2562, 19236, 148085, 1167488, 9409645, 77367087, ...
1, 8, 64, 520, 4304, 36320, 312512, 2740672, 24476800, 222358528, ...
1, 9, 81, 738, 6822, 64026, 610245, 5906502, 58033953, 578488563, ...
1, 10, 100, 1010, 10320, 106740, 1117880, 11855660, 127313320, ...
Successive self-compositions of F(x), the g.f. of A120010, begin:
F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + 3x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 68x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + 710x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + 9348x^6 +...

Rows: A120010, A120017, A120018; Diagonals: A120020, A120021. Variant: A120013.

{T(n,k)=sum(j=1, k, binomial(2*k-2*j, k-j)/(k-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(k-j+i, j-i)*binomial(k-j+i-1, i-1)*n^(i-1)))}

T(n, k) = Sum_{j=1..k}Catalan(k-j)*[Sum_{i=1..j}(-1)^(j-i)*n^(i-1)*C(k-j+i, j-i)*C(k-j+i-1, i-1)]; Also, T(n, k) = Sum_{j=0..k-1}n^j*[Sum_{i=j..k-1}(-1)^(i-j)*Catalan(k-i-1)*C(k-i+j, i-j)*C(k-i+j-1, j)]; where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

A120022 a(n) = A120020(n)/n = coefficient of x^n in the n-th self-composition of the g.f. of A120010, divided by n, for n>=1.

Original entry on oeis.org

1, 1, 3, 17, 142, 1558, 21155, 342584, 6448217, 138392304, 3336869488, 89325958048, 2629214627421, 84408934941424, 2935694381925743, 109967573757472768, 4414292541216287516, 189054708982869449056

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

Cf. A120010, A120019, A120020, A120021, A120016.

a[n_] := Sum[((-1)^(j-i) n^(i-2) Binomial[2n-2j, n-j] Binomial[n+i-j, j-i] Binomial[n+i-j-1, i-1])/(n-j+1), {j, 1, n}, {i, 1, j}]; Array[a, 18] (* Jean-François Alcover, Nov 14 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x*(1-x)/(1-(n+1)*x*(1-x)+x*O(x^n))))/2, n)/n

/* Alternate Formula: */ a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-2)))

cp's OEIS Frontend

A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4x))/2 o x/(1-nx) o (x-x^2) } for n>=1.

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A120019 Square table, read by antidiagonals, of self-compositions of A120010.

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A120022 a(n) = A120020(n)/n = coefficient of x^n in the n-th self-composition of the g.f. of A120010, divided by n, for n>=1.

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cp's OEIS Frontend

A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) } for n>=1.

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A120019 Square table, read by antidiagonals, of self-compositions of A120010.

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A120022 a(n) = A120020(n)/n = coefficient of x^n in the n-th self-composition of the g.f. of A120010, divided by n, for n>=1.

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A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4x))/2 o x/(1-nx) o (x-x^2) } for n>=1.