cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A026000 a(n) = T(2n, n), where T is the Delannoy triangle (A008288).

Original entry on oeis.org

1, 5, 41, 377, 3649, 36365, 369305, 3800305, 39490049, 413442773, 4354393801, 46082942185, 489658242241, 5220495115997, 55818956905529, 598318746037217, 6427269150511105, 69175175263888037, 745778857519239785, 8052432236270744665, 87063177396677721409
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Even order terms in the diagonal of rational function 1/(1 - (x + y^2 + x*y^2)). - Gheorghe Coserea, Aug 31 2018

Examples

			A(x) = 1 + 5*x + 41*x^2 + 377*x^3 + 3649*x^4 + 36365*x^5 + ...

Links

Crossrefs

Cf. A008288, A027307, A026001.

Programs

  • Mathematica
    Flatten[{1,RecurrenceTable[{2*n*(2*n-1)*a[n] == (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3],a[1]==5,a[2]==41,a[3]==377},a,{n,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[n_] :=  HypergeometricPFQ[{-n, -n, n + 1}, {1/2, 1}, 1];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 14 2018 *)
  • PARI
    seq(N) = {
  my(a = vector(N)); a[1]=5; a[2]=41; a[3]=377;
  for (n=4, N,
    a[n] = (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3];
    a[n] /= 2*n*(2*n-1));
  concat(1, a);
};
seq(18)
\\ test: y=Ser(seq(303),'x); 0 == 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1
\\ Gheorghe Coserea, Aug 31 2018

Formula

a(n) = ((2*n+3)*(n+1)*A027307(n+1)/2-(3*n+2)*n*A027307(n)) / (5*n+3) (guessed). - Mark van Hoeij, Jul 02 2010
Recurrence: 2*n*(2*n-1)*a(n) = (46*n^2-51*n+15)*a(n-1) - (18*n^2-82*n+85)*a(n-2) - (n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(150+70*sqrt(5))*((11+5*sqrt(5))/2)^n/(20*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012. Equivalently, a(n) ~ phi^(5*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
a(n) = hypergeom([-n, -n, n + 1], [1/2, 1], 1). - Peter Luschny, Mar 14 2018
From Gheorghe Coserea, Aug 31 2018:(Start)
G.f.: 1 + serreverse((-(44*x^2 + 88*x + 45) + (10*x + 9)*sqrt(20*x^2 + 44*x + 25))/(8*(x + 1)^2)).
G.f. y=A(x) satisfies:
0 = 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1.
0 = 2*x*(x - 2)*(x^2 + 11*x - 1)*y'' + (5*x^3 + 8*x^2 - 87*x + 2)*y' + (x^2 - 7*x - 10)*y. (End)
From Peter Bala, Jan 20 2020: (Start)
a(n) = Sum_{k = 0..n} C(2*n, n-k) * C(2*n+k, k).
a(n) = C(2*n, n) * hypergeom([-n, 2*n+1], [n+1], -1).
n*(2*n-1)*(10*n-13)*a(n) = (220*n^3-506*n^2+334*n-63*n)*a(n-1) + (n-1)*(2*n-3)*(10*n-3)*a(n-2). (End)
From Peter Bala, Apr 15 2023: (Start)
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(2*n, k)*2^k
a(n) = (-1)^n * Sum_{k = 0..n} binomial(n, k)*binomial(2*n+k, k)*(-2)^k.
a(n) = hypergeom([-n, -2*n], [1], 2) = (-1)^n * hypergeom([-n, 2*n + 1], [1], 2). (End)

A331329 a(n) = binomial(5*n, n)*hypergeom([-4*n, -n], [-5*n], -1).

Original entry on oeis.org

1, 9, 145, 2625, 50049, 982729, 19665841, 398796225, 8166636545, 168502295625, 3497529199185, 72949645000065, 1527671538372225, 32100078290806665, 676451066002195825, 14290577765009652865, 302557549412667613185, 6417968867896642617225, 136371773642235542394385
Offset: 0

Views

Author

Peter Luschny, Jan 31 2020

Keywords

Comments

Special case of generalized Delannoy numbers (see cross-references):
T(n, k) = binomial(k*n, n)*hypergeom([(1-k)*n, -n], [-k*n], -1).

Links

Crossrefs

Cf. A001850 (k=2), A026000 (k=3), A026001 (k=4), this sequence (k=5), A341491 (k=6).
Cf. A008288, A181675, A341476, A341477.

Programs

  • Mathematica
    a[n_] := Binomial[5 n, n] Hypergeometric2F1[-4 n, -n, -5 n, -1];
Array[a, 19, 0]

Formula

a(n) ~ sqrt(5 + 21/sqrt(17)) * (349 + 85*sqrt(17))^n / (sqrt(Pi*n) * 2^(5*n + 2)). - Vaclav Kotesovec, Feb 13 2021

A341491 a(n) = binomial(6*n, n) * hypergeom([-5*n, -n], [-6*n], -1).

Original entry on oeis.org

1, 11, 221, 4991, 118721, 2908411, 72616013, 1837271615, 46943003137, 1208483403179, 31297149356221, 814471993937855, 21281058718583873, 557930580979801755, 14669716953106628781, 386675596518995000191, 10214494658006725840897, 270345191656309313382475
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 13 2021

Keywords

Comments

In general, for k > 1, binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1) ~ sqrt((k + (k^2 - k + 1) / sqrt(k^2 - 2*k + 2)) / (4*(k-1)*Pi*n)) * ((A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1))^n.

Links

Crossrefs

Cf. A001850, A026000, A026001, A331329, A341476, A341477.

Programs

  • Mathematica
    Table[Binomial[6*n, n] * Hypergeometric2F1[-5*n, -n, -6*n, -1], {n,0,20}]

Formula

a(n) ~ sqrt((6 + 31/sqrt(26))/(20*Pi*n)) * (42671 + 8346*sqrt(26))^n / 5^(5*n).

A341476 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

1, 3, 22, 223, 2792, 42671, 761984, 15707707, 365122688, 9491746747, 271962399232, 8539383210711, 290937486190592, 10710312199270503, 422984587596455936, 17864076455770831219, 802450164859200372736, 38242916911507537149427, 1925477163696152909447168, 102213291475268656299164879
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 13 2021

Keywords

Examples

			Lim_{n->infinity} A001850(n)^(1/n) = (    3 +    2 * sqrt(1^2 + 1)) / 1^1.
Lim_{n->infinity} A026000(n)^(1/n) = (   22 +   10 * sqrt(2^2 + 1)) / 2^2.
Lim_{n->infinity} A026001(n)^(1/n) = (  223 +   70 * sqrt(3^2 + 1)) / 3^3.
Lim_{n->infinity} A331329(n)^(1/n) = ( 2792 +  680 * sqrt(4^2 + 1)) / 4^4.
Lim_{n->infinity} A341491(n)^(1/n) = (42671 + 8346 * sqrt(5^2 + 1)) / 5^5.

Links

Crossrefs

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341477.

Formula

Lim_{n->infinity} (binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1))^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1), for k>1.
Lim_{n->infinity} hypergeom([(1-k)*n, -n], [-k*n], -1)^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / k^k.
For k > 1, A341476(k)^2 - ((k-1)^2 + 1) * A341477(k)^2 = (-1)^k * (k-1)^(2*k-2).
Lim_{k->infinity} (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k * (k-1)^(k-1)) = 2*exp(1).
a(n) ~ n^n.

A341477 Coefficients related to the asymptotics of generalized Delannoy numbers.

Original entry on oeis.org

0, 2, 10, 70, 680, 8346, 125504, 2218350, 45335680, 1047314578, 27079557632, 772687787510, 24172386314240, 821114930966890, 30146801401143296, 1187943632192716894, 50068690149298438144, 2245175953053786221730, 106828553482726336102400, 5371204894269759411503910
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 13 2021

Keywords

Examples

			Lim_{n->infinity} A001850(n)^(1/n) = (    3 +    2 * sqrt(1^2 + 1)) / 1^1.
Lim_{n->infinity} A026000(n)^(1/n) = (   22 +   10 * sqrt(2^2 + 1)) / 2^2.
Lim_{n->infinity} A026001(n)^(1/n) = (  223 +   70 * sqrt(3^2 + 1)) / 3^3.
Lim_{n->infinity} A331329(n)^(1/n) = ( 2792 +  680 * sqrt(4^2 + 1)) / 4^4.
Lim_{n->infinity} A341491(n)^(1/n) = (42671 + 8346 * sqrt(5^2 + 1)) / 5^5.

Links

Crossrefs

Cf. A008288, A001850, A026000, A026001, A331329, A341491, A341476.

Formula

Lim_{n->infinity} (binomial(k*n, n) * hypergeom([(1-k)*n, -n], [-k*n], -1))^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k-1)^(k-1), for k>1.
Lim_{n->infinity} hypergeom([(1-k)*n, -n], [-k*n], -1)^(1/n) = (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / k^k.
For k > 1, A341476(k)^2 - ((k-1)^2 + 1) * A341477(k)^2 = (-1)^k * (k-1)^(2*k-2).
Lim_{k->infinity} (A341476(k) + A341477(k)*sqrt((k-1)^2 + 1)) / (k * (k-1)^(k-1)) = 2*exp(1).
a(n) ~ n^(n-1).

A181675 V(n,n^2), where V is the number of integer points in an n-dimensional sphere of Lee-radius n^2 centered at the origin.

Original entry on oeis.org

3, 41, 1159, 50049, 2908411, 212358985, 18665359119, 1917971421185, 225555471838387, 29871434052884841, 4398867465890529303, 712959801840558794625, 126115813138335816685995
Offset: 2

Views

Author

Antonio Campello, Nov 04 2010

Keywords

Comments

Since V(n,d) is symmetric, we have V(n,n^2) = V(n^2,n).

Links

Crossrefs

V(n, n) = A001850, V(n, 2n) = A026000 and V(n, 3n) = A026001.

Programs

  • Mathematica
    Array[Sum[2^j*Binomial[#1, j] Binomial[#2, j], {j, 0, Min[#1, #2]}] & @@ {#, #^2} &, 13] (* Michael De Vlieger, Jul 05 2019 *)

Formula

V(n,d) = Sum_{j=0..min(n,d)} 2^j * binomial(n,j)*binomial(d,j).
a(n) ~ exp(n-2) * (2*n)^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 13 2021

A227964 Triangle where the g.f. of row n equals (1-x-x^2+x^3)^n and terms T(n,k) are read by rows n>=0, k=0..3*n.

Original entry on oeis.org

1, 1, -1, -1, 1, 1, -2, -1, 4, -1, -2, 1, 1, -3, 0, 8, -6, -6, 8, 0, -3, 1, 1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1, 1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1, 1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1, 1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2013

Keywords

Examples

			Triangle begins:
1;
1, -1, -1, 1;
1, -2, -1, 4, -1, -2, 1;
1, -3, 0, 8, -6, -6, 8, 0, -3, 1;
1, -4, 2, 12, -17, -8, 28, -8, -17, 12, 2, -4, 1;
1, -5, 5, 15, -35, -1, 65, -45, -45, 65, -1, -35, 15, 5, -5, 1;
1, -6, 9, 16, -60, 24, 116, -144, -66, 220, -66, -144, 116, 24, -60, 16, 9, -6, 1;
1, -7, 14, 14, -91, 77, 168, -344, -14, 546, -364, -364, 546, -14, -344, 168, 77, -91, 14, 14, -7, 1; ...

Links

Crossrefs

Cf. A192205.

Programs

  • PARI
    {T(n,k)=polcoeff((1-x-x^2+x^3 +x*O(x^k))^n,k)}
for(n=0,10,for(k=0,3*n,print1(T(n,k),", "));print(""))

Formula

Sum_{k=0..3*n} |T(n,k)| = A192205(n).
Sum_{k=0..3*n} T(n,k)^2 = binomial(4*n,n).
Sum_{k=0..3*n} T(n,k) * binomial(3*n,k) = (-1)^n * binomial(4*n,n).
Sum_{k=0..3*n} T(n,k) * binomial(2*n+k,k) = 2^n.
Sum_{k=0..3*n} T(n,k) * binomial(3*n+k,k) = A008288(3*n,n), where A008288 is the Delannoy array (see A026001).

A341470 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 41, 63, 1, 1, 9, 85, 377, 321, 1, 1, 11, 145, 1159, 3649, 1683, 1, 1, 13, 221, 2625, 16641, 36365, 8989, 1, 1, 15, 313, 4991, 50049, 246047, 369305, 48639, 1, 1, 17, 421, 8473, 118721, 982729, 3707509, 3800305, 265729, 1
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2021

Keywords

Examples

			Square array begins:
  1,    1,     1,      1,      1,       1, ...
  1,    3,     5,      7,      9,      11, ...
  1,   13,    41,     85,    145,     221, ...
  1,   63,   377,   1159,   2625,    4991, ...
  1,  321,  3649,  16641,  50049,  118721, ...
  1, 1683, 36365, 246047, 982729, 2908411, ...

Links

Crossrefs

Columns k=0..5 give A000012, A001850, A026000, A026001, A331329, A341491.
Rows n=0..2 give A000012, A005408, A102083.
Main diagonal gives A181675(n+1).
Cf. A008288.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(k*n, n-j)*binomial(k*n+j, j));
  • PARI
    T(n, k) = sum(j=0, n, 2^j*binomial(n, j)*binomial(k*n, j));

Formula

T(n,k) = A008288(n,k*n).
T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j) * binomial(k*n,j).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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