A086619 -id:A086619 - cp's OEIS frontend

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 79, 47, 11, 1, 1, 13, 71, 175, 175, 71, 13, 1, 1, 15, 100, 331, 504, 331, 100, 15, 1, 1, 17, 134, 562, 1196, 1196, 562, 134, 17, 1, 1, 19, 173, 883, 2464, 3514, 2464, 883, 173, 19, 1, 1, 21, 217

Offset: 0

Paul D. Hanna, Jul 24 2003

Determinants of upper left n X n matrices results in A086619: {1,2,10,150,7650,1438200,1051324200,...}, which is the products of the first n terms of the binomial transform of Catalan numbers (A007317): {1,2,5,15,51,188,731,2950,...}.

			Rows begin:
1,_1,__1,__1,___1,____1,____1,_____1, ...
1,_3,__5,__7,___9,___11,___13,____15, ...
1,_5,_14,_28,__47,___71,__100,___134, ...
1,_7,_28,_79,_175,__331,__562,___883, ...
1,_9,_47,175,_504,_1196,_2464,__4572, ...
1,11,_71,331,1196,_3514,_8764,_19244, ...
1,13,100,562,2464,_8764,26172,_67740, ...
1,15,134,883,4572,19244,67740,204831, ...

Cf. A086621 (diagonal), A086622 (antidiagonal sums), A086619 (determinants).

Contribution from Paul Barry, Feb 04 2009: (Start)
T(n,k)=sum{j=0..n+k, C(k,j-k)*C(n+2k-j,k)*if(k<=j,A000108(n-k),0)};
Regarded as a number triangle read by row, columns are generated by sum{j=0..k, C(k,j)*A000108(j)*x^j}*x^k/(1-x)^(k+1). (End)

A294352 Product of first n terms of the binomial transform of the factorial.

Original entry on oeis.org

1, 2, 10, 160, 10400, 3390400, 6635012800, 90899675360000, 9962695319131360000, 9827302289744364817600000, 96937502343569678741652977600000, 10518214548789290471667075399621491200000, 13695360582395151673134516587047571322777664000000

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..43
M. Bahrami-Taghanaki, A. R. Moghaddamfar, Nima Salehy, and Navid Salehy, Some Identities Involving Stirling Numbers Arising from Matrix Decompositions, J. Int. Seq. (2024) Vol. 24, No. 5, Art. No. 24.5.3. See p. 9.

Cf. A000522, A047656, A086619, A294353.

Table[Product[Sum[Binomial[m, k]*k!, {k, 0, m}], {m, 0, n}], {n, 0, 12}]

a(n) ~ c * exp(n+1) * BarnesG(n+2).
a(n) ~ c * n^(n^2/2 + n + 5/12) * (2*Pi)^(n/2 + 1/2) / (A * exp(3*n^2/4 - 13/12))
where c = 0.24314714161123874545254157058990661627416712475691705561000082745...
and A is the Glaisher-Kinkelin constant A074962.

A294353 Product of first n terms of the binomial transform of n^n (A086331).

Original entry on oeis.org

1, 2, 14, 602, 236586, 1116922506, 78020387811618, 95634036502805444826, 2378081951650318040462277306, 1361239109900199746154166909875717978, 20062823024247092576000017563809908231829439138, 8420023655209092490508999978430595224656730339006712229850

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Cf. A002109, A047656, A086331, A086619, A294352.

Table[Product[1 + Sum[Binomial[m, k]*k^k, {k, 1, m}], {m, 0, n}], {n, 0, 12}]

a(n) ~ c * n^(n*(n+1)/2 + 1/12 + exp(-1)/2) / exp(n^2/4 - n*exp(-1)), where c = 1.981849007720509372587479129359338230641860983165241915197508762536...

A294350 Product of first n terms of the binomial transform of the partition function (A000041).

Original entry on oeis.org

1, 2, 10, 130, 4420, 388960, 87516000, 49796604000, 70960160700000, 251057048556600000, 2188464292267882200000, 46682131818366195208200000, 2421822316605019841206207800000, 303875733353698259555507717497200000, 91748896295748761809334889636212098800000

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Cf. A086619, A218481, A294351.

Table[Product[Sum[Binomial[m, k]*PartitionsP[k], {k, 0, m}], {m, 0, n}], {n, 0, 15}]

a(n) = prod(m=0, n, sum(k=0, m, binomial(m,k)*numbpart(k))); \\ Michel Marcus, Oct 29 2017

A294351 Product of first n terms of the binomial transform of the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 2, 8, 72, 1512, 74088, 8446032, 2238198480, 1376492065200, 1957371716714400, 6404520257089516800, 47989070286371749382400, 820133211194093196945216000, 31862175254890520701321641600000, 2805942463821933705561890367504000000

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Cf. A086619, A266232, A294350.

Table[Product[Sum[Binomial[m, k]*PartitionsQ[k], {k, 0, m}], {m, 0, n}], {n, 0, 15}]

A102318 A mean binomial transform of the Catalan numbers.

Original entry on oeis.org

1, 1, 3, 8, 27, 97, 373, 1493, 6163, 26027, 111897, 488006, 2153429, 9596199, 43121211, 195165576, 888861555, 4070582971, 18732710281, 86584519280, 401776434017, 1870983991035, 8740907398527, 40956401225597

Offset: 0

Paul Barry, Jan 04 2005

Average of binomial and inverse binomial transforms of the Catalan numbers A000108.

Cf. A007317, A086619.

G.f.: (2-sqrt((1-3x)/(1+x))-sqrt((1-5x)/(1-x)))/(4x);
a(n)=sum{k=0..floor(n/2), binomial(n, 2k)C(n-2k)};
a(n)=sum{k=0..n, binomial(n, k)C(k)(1+(-1)^(n-k))/2}.
Conjecture: -(n-1)*(n+1)*a(n) +2*(5*n^2-9*n+1)*a(n-1) +2*(-15*n^2+58*n-49)*a(n-2) +2*(10*n^2-76*n+123)*a(n-3) +(31*n-55)*(n-3)*a(n-4) -30*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 08 2016
Conjecture: +(3*n-10)*(n-1)*(n+1)*a(n) +2*(-12*n^3+58*n^2-67*n+10)*a(n-1) +2*(21*n^3-136*n^2+289*n-196)*a(n-2) +2*(n-2)*(12*n^2-46*n+27)*a(n-3) -15*(n-2)*(n-3)*(3*n-7)*a(n-4)=0. - R. J. Mathar, Jun 08 2016
a(n) ~ 5^(n + 3/2) / (16 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 30 2017

A294349 Product of first n terms of the binomial transform of the Lucas numbers (A000032).

Original entry on oeis.org

2, 6, 42, 756, 35532, 4370436, 1407280392, 1186337370456, 2618246576596392, 15128228719573952976, 228844715840995186667952, 9062937281450932377610903056, 939663463215395570775453650652192, 255065069445576619918001465293982953056

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Cf. A005248, A047656, A086619, A135407, A194157.

Table[Product[Sum[Binomial[m, k]*LucasL[k], {k, 0, m}], {m, 0, n}], {n, 0, 15}]
Table[Product[LucasL[2*k], {k, 0, n}], {n, 0, 15}]

a(n) ~ c * phi^(n*(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 2.349094356918735309421297337651771419003525539652230102934874983942...

cp's OEIS Frontend

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^2.

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A294352 Product of first n terms of the binomial transform of the factorial.

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A294353 Product of first n terms of the binomial transform of n^n (A086331).

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A294350 Product of first n terms of the binomial transform of the partition function (A000041).

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A294351 Product of first n terms of the binomial transform of the number of partitions into distinct parts (A000009).

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A102318 A mean binomial transform of the Catalan numbers.

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A294349 Product of first n terms of the binomial transform of the Lucas numbers (A000032).

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Formula

A086620 Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^ny^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xyf(x,y)^2.