A101514 -id:A101514 - cp's OEIS frontend

A101515 Symmetric square array, read by antidiagonals, such that the inverse binomial transform of row n forms the sequence: {C(n,k)*A101514(k), 0<=k<=n}, where A101514 equals the main diagonal shift right.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 35, 21, 6, 1, 1, 7, 31, 77, 77, 31, 7, 1, 1, 8, 43, 146, 236, 146, 43, 8, 1, 1, 9, 57, 249, 596, 596, 249, 57, 9, 1, 1, 10, 73, 393, 1290, 2037, 1290, 393, 73, 10, 1, 1, 11, 91, 585, 2486, 5772, 5772, 2486, 585

Offset: 0

Paul D. Hanna, Dec 06 2004

The main diagonal equals A101514 shift one place left. The antidiagonal sums form A101516.

			Rows begin:
[_1,1,1,1,1,1,1,1,1,...],
[1,_2,3,4,5,6,7,8,9,...],
[1,3,_7,13,21,31,43,57,73,...],
[1,4,13,_35,77,146,249,393,585,...],
[1,5,21,77,_236,596,1290,2486,4387,...],
[1,6,31,146,596,_2037,5772,13987,29987,...],
[1,7,43,249,1290,5772,_21695,67943,181811,...],
[1,8,57,393,2486,13987,67943,_277966,951051,...],
[1,9,73,585,4387,29987,181811,951051,_4198635,...],...
The inverse binomial transform of the rows of this array are generated
from the products of the main diagonal with rows of Pascal's triangle:
BINOMIAL[1*1] = [_1,1,1,1,1,1,1,1,1,...],
BINOMIAL[1*1,1*1] = [1,_2,3,4,5,6,7,8,9,...],
BINOMIAL[1*1,1*2,2*1] = [1,3,_7,13,21,31,43,57,73,...],
BINOMIAL[1*1,1*3,2*3,7*1] = [1,4,13,_35,77,146,249,393,...],
BINOMIAL[1*1,1*4,2*6,7*4,35*1] = [1,5,21,77,_236,596,1290,...],
BINOMIAL[1*1,1*5,2*10,7*10,35*5,236*1] = [1,6,31,146,596,_2037,...],...

Cf. A101514, A101516.

T(n,k)=if(n<0 || k<0,0,if(n==0 || k==0,1,if(n>k,T(k,n), 1+sum(j=1,k,binomial(k,j)*binomial(n,j)*T(j-1,j-1));)))

A101516 Antidiagonal sums of symmetric square array A101515 and also equals the binomial transform of a sequence formed from terms of A101514 repeated twice.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 91, 232, 632, 1824, 5571, 17892, 60355, 212898, 784416, 3008480, 11997341, 49612426, 212536067, 941213428, 4305049140, 20302469824, 98641434683, 493038167880, 2533414749409, 13366134856170, 72361098996208

Offset: 0

Paul D. Hanna, Dec 06 2004

nonn

A101514 equals the main diagonal of A101515 shift one place right and also A101514 shifts one place left under the square binomial transform (A008459): A101514(n+1) = Sum_{k=0..n-1} C(n-1,k)^2*A101514(k).

			Given A101514 = [1,1,2,7,35,236,2037,21695,277966,4198635,...],
the binomial transform of A101514 terms repeated twice returns this sequence:
BINOMIAL[1,1,1,1,2,2,7,7,35,35,...] = [1,2,4,8,17,38,91,232,632,1824,...].

Cf. A101514, A101515.

{a(n)=sum(k=0,n,binomial(n,k)* if(k\2==0,1,sum(j=0,k\2-1,binomial(k\2-1,j)^2* sum(i=0,2*j,(-1)^(2*j-i)*binomial(2*j,i)*a(i)))))}

G.f.: A(x) = G101514(x^2/(1-x)^2)/(1-x)^2, where G101514(x)= g.f. of A101514. a(n) = Sum_{k=0..n} C(n, k)*A101514([k/2]).

A342196 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^2 * a(k-1).

Original entry on oeis.org

1, 1, 5, 23, 155, 1355, 14371, 183911, 2781283, 48726355, 976903875, 22183097191, 565060532965, 16016170519017, 501714014484813, 17265124180702953, 649178961366102597, 26544344366333824055, 1175291769917975444817, 56133021061270139242637, 2881893164859601701738005

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A023998, A040027, A101514, A102221, A342197, A342198, A342199.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]

A346185 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * a(k).

Original entry on oeis.org

1, 1, 2, 11, 93, 1294, 26045, 714391, 26109426, 1224739755, 71807248783, 5173027197636, 450173748220033, 46617339568635115, 5677430539873463470, 804907754967314483801, 131598260940217897338131, 24609634809861999705338820, 5226508081059269450476666513

Offset: 0

Ilya Gutkovskiy, Jul 09 2021

nonn

Cf. A000110, A061684, A101514, A181543, A336195.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^3 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]

A342182 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - x * BesselI(0,2*sqrt(x))).

Original entry on oeis.org

1, 1, 8, 117, 3184, 134025, 8141436, 672837277, 72634878016, 9923765772177, 1673881314096700, 341631408064928421, 82978986493779894288, 23653894531273155603961, 7819996460332550715977588, 2967815528758036870644773925, 1281517958938232539844046259456

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A006153, A101514, A336228, A337591.

nmax = 16; CoefficientList[Series[1/(1 - x BesselI[0, 2 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(Binomial[n, k] (n - k))^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

a(n) = {n!^2*polcoef(1 / (1 - sum(k=1, n, x^k / ((k-1)!)^2) + O(x*x^n)), n)} \\ Andrew Howroyd, Mar 04 2021

Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - Sum_{n>=1} x^n / ((n-1)!)^2).
a(0) = 1; a(n) = Sum_{k=0..n-1} (binomial(n,k) * (n-k))^2 * a(k).
a(n) ~ n!^2 / ((1 + r^(3/2)*BesselI(1, 2*sqrt(r))) * r^n), where r = 0.592860029867912878114616561736048937618032595935338954527835... is the root of the equation r*BesselI(0, 2*sqrt(r)) = 1. - Vaclav Kotesovec, May 04 2024

cp's OEIS Frontend

A101515 Symmetric square array, read by antidiagonals, such that the inverse binomial transform of row n forms the sequence: {C(n,k)*A101514(k), 0<=k<=n}, where A101514 equals the main diagonal shift right.

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PARI

A101516 Antidiagonal sums of symmetric square array A101515 and also equals the binomial transform of a sequence formed from terms of A101514 repeated twice.

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PARI

Formula

A342196 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^2 * a(k-1).

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Mathematica

A346185 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * a(k).

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Mathematica

A342182 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - x * BesselI(0,2*sqrt(x))).

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Mathematica

PARI

Formula