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 1-9 of 9 results.

A355722 Row 2 of table A355721.

Original entry on oeis.org

1, 2, 14, 138, 1686, 24162, 394254, 7191018, 144786006, 3188449602, 76246683534, 1968284351178, 54576250392726, 1618348891438242, 51122453577462414, 1714406473587300138, 60843580566100937046, 2278637898592632599682, 89818339421620249242894, 3717488491001699691500298
Offset: 0

Views

Author

Peter Bala, Jul 15 2022

Keywords

Links

Crossrefs

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355723 (row 3), A355724 (row 4), A355725 (row 5).

Programs

  • Maple
    n := 2: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f: A(x) = ( Sum_{k >= 0} d(k+2)/d(2)*x^k )/( Sum_{k >= 0} d(k+1)/d(1)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x)= 1/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 3*x*A(x)^2 - (1 + x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - 9*x/(1 - ... - 2*n*x/(1 - (2*n+3)*x )))))))), a continued fraction of Stieltjes type.

A355723 Row 3 of table A355721.

Original entry on oeis.org

1, 2, 18, 218, 3194, 53890, 1019250, 21256090, 483426010, 11895873410, 314834663250, 8918883839450, 269367643864250, 8643467766472450, 293770652998691250, 10546424484691428250, 398914704362503668250, 15860639479547463637250, 661439858772303085871250, 28874834455755565593004250
Offset: 0

Views

Author

Peter Bala, Jul 15 2022

Keywords

Links

Crossrefs

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355724 (row 4), A355725 (row 5).

Programs

  • Maple
    n := 3: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f: A(x) = ( Sum_{k >= 0} d(k+3)/d(3)*x^k )/( Sum_{k >= 0} d(k+2)/d(2)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x) = 1/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 5*x*A(x)^2 - (1 + 3*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 9*x/(1 - 6*x/(1 - 11*x/(1 - ... - 2*n*x/(1 - (2*n+5)*x )))))))), a continued fraction of Stieltjes type.

A355724 Row 4 of table A355721.

Original entry on oeis.org

1, 2, 22, 314, 5326, 102722, 2197558, 51355514, 1297759918, 35208930050, 1020115715542, 31432396066106, 1026506419425550, 35428218801977666, 1288967076156307702, 49323199246104202874, 1980947315202528449518, 83342865788161594337282, 3666525676611059535630742
Offset: 0

Views

Author

Peter Bala, Jul 15 2022

Keywords

Links

Crossrefs

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355725 (row 5).

Programs

  • Maple
    n := 4: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f: A(x) = ( Sum_{k >= 0} d(k+4)/d(4)*x^k )/( Sum_{k >= 0} d(k+3)/d(3)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x) = 1/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - 13*x/(1 + 13*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 7*x*A(x)^2 - (1 + 5*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - 13*x/(1 - ... - 2*n*x/(1 - (2*n+7)*x )))))))), a continued fraction of Stieltjes-type.

A355725 Row 5 of table A355721.

Original entry on oeis.org

1, 2, 26, 426, 8178, 176802, 4206618, 108577674, 3011332338, 89141101506, 2802596567706, 93232011912426, 3271729161905010, 120810104634555234, 4683805718871051162, 190294015841923438026, 8087576641287426829170, 358981130096398432055682, 16615841072836741527510810
Offset: 0

Views

Author

Peter Bala, Jul 15 2022

Keywords

Links

Crossrefs

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355724 (row 4).

Programs

  • Maple
    n := 5: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f: A(x) = ( Sum_{k >= 0} d(k+5)/d(5)*x^k )/( Sum_{k >= 0} d(k+4)/d(4)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x) = 1/(1 + 9*x - 11*x/(1 + 11*x - 13*x/(1 + 13*x - 15*x/(1 + 15*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 9*x*A(x)^2 - (1 + 7*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - 15*x/(1 - ... - 2*n*x/(1 - (2*n+9)*x )))))))), a continued fraction of Stieltjes-type.

A355793 Square table, read by antidiagonals: the g.f. for row n is given recursively by (3*n-1)*x*R(n,x) = 1 + (3*n-4)*x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112936(k+1)*x^k.

Original entry on oeis.org

1, 1, 3, 1, 3, 15, 1, 3, 24, 111, 1, 3, 33, 282, 1131, 1, 3, 42, 507, 4236, 14943, 1, 3, 51, 786, 9609, 76548, 243915, 1, 3, 60, 1119, 17736, 212835, 1608864, 4742391, 1, 3, 69, 1506, 29103, 459768, 5350785, 38488152, 106912131, 1, 3, 78, 1947, 44196, 859143, 13333488
Offset: 0

Views

Author

Peter Bala, Jul 17 2022

Keywords

Comments

Compare with A111528 and A355721, which have similar definitions and properties.

Examples

			Square array begins
1, 3, 15,  111,  1131,   14943,    243915,    4742391,    106912131, ...
1, 3, 24,  282,  4236,   76548,   1608864,   38488152,   1032125136, ...
1, 3, 33,  507,  9609,  212835,   5350785,  149961675,   4628365305, ...
1, 3, 42,  786, 17736,  459768,  13333488,  425600976,  14791250688, ...
1, 3, 51, 1119, 29103,  859143,  28091463, 1002057591,  38606468343, ...
1, 3, 60, 1506, 44196, 1458588,  52917360, 2080630776,  87823112496, ...
1, 3, 69, 1947, 63501, 2311563,  91949469, 3943276347, 180679742061, ...
1, 3, 78, 2442, 87504, 3477360, 150259200, 6970190160, 344116224960, ...

Links

Crossrefs

Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355796 (row 3), A355797 (row 4). Cf. A008544, A111528, A355721.

Programs

  • Maple
    T := (n,k) -> coeff(series(hypergeom([n+2/3, 1], [], 3*x)/ hypergeom([n-1/3, 1], [], 3*x), x, 21), x, k):
# display as a sequence
seq(seq(T(n-k,k), k = 0..n), n = 0..10);
# display as a square array
seq(print(seq(T(n,k), k = 0..10)), n = 0..10);

Formula

Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
O.g.f. for row n >= 0: R(n,x) = ( Sum_{k >= 0} t(n+k)/t(n)*x^k )/( Sum_{k >= 0} t(n-1+k)/t(n-1)*x^k ).
R(n,x)/(1 - (3*n-1)*x*R(n,x)) = Sum_{k >= 0} t(n+k)/t(n)*x^k.
R(n,x) = 1/(1 + (3*n-1)*x - (3*n+2)*x/(1 + (3*n+2)*x - (3*n+5)*x/(1 + (3*n+5)*x - (3*n+8)*x/(1 + (3*n+8)*x - ... )))) (continued fraction).
R(n,x) satisfies the Riccati differential equation 3*x^2*d/dx(R(n,x)) + (3*n-1)*x*R(n,x)^2 - (1 + (3*n-4)*x)*R(n,x) + 1 = 0 with R(n,0) = 1.
Applying Stokes 1982 gives R(n,x) = 1/(1 - 3*x/(1 - (3*n+2)*x/(1 - 6*x/(1 - (3*n+5)*x/(1 - 9*x/(1 - (3*n+8)*x/(1 - 12*x/(1 - ...)))))))), a continued fraction of Stieltjes type.

A355794 Row 1 of A355793.

Original entry on oeis.org

1, 3, 24, 282, 4236, 76548, 1608864, 38488152, 1032125136, 30670171248, 1000637672064, 35571839009952, 1368990872569536, 56720594992438848, 2517761078627172864, 119222916630934484352, 5999613754698100628736, 319763269764299852744448, 17994913747767982690289664
Offset: 0

Views

Author

Peter Bala, Jul 19 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355795 (row 2), A355796 (row 3), A355797 (row 4).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 1: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f.: A(x) = ( Sum_{k >= 0} t(k+1)/t(1)*x^k )/( Sum_{k >= 0} t(k)/t(0)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
A(x)/(1 - 2*x*A(x)) = Sum_{k >= 0} t(k+1)/t(1)*x^k.
A(x) = 1/(1 + 2*x - 5*x/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*A(x)' + 2*x*A(x)^2 - (1 - x)*A(x) + 1 = 0 with A(0) = 1.
Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 5*x/(1 - 6*x/(1 - 8*x/(1 - 9*x/(1 - 11*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.

A355795 Row 2 of A355793.

Original entry on oeis.org

1, 3, 33, 507, 9609, 212835, 5350785, 149961675, 4628365305, 155913036915, 5692874399025, 224034935130075, 9456933847187625, 426402330032719875, 20460268520575152225, 1041301103429870128875, 56040353252589013121625, 3180443637298592493577875, 189863589771186976073108625
Offset: 0

Views

Author

Peter Bala, Jul 21 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355794 (row 1), A355796 (row 3), A355797 (row 4).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 2: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f.: A(x) = ( Sum_{k >= 0} t(k+2)/t(2)*x^k )/( Sum_{k >= 0} t(k+1)/t(1)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
A(x)/(1 - 5*x*A(x)) = Sum_{k >= 0} t(k+2)/t(2)*x^k.
A(x) = 1/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*A(x)' + 5*x*A(x)^2 - (1 + 2*x)*A(x) + 1 = 0 with A(0) = 1.
Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 8*x/(1 - 6*x/(1 - 11*x/(1 - 9*x/(1 - 14*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.

A355796 Row 3 of A355793.

Original entry on oeis.org

1, 3, 42, 786, 17736, 459768, 13333488, 425600976, 14791250688, 555381292800, 22398626084352, 965768866650624, 44347055502428160, 2161455366606034944, 111489317304231616512, 6069676735484389779456, 347921629212782938472448, 20950823605616500202323968, 1322561808699778749456678912
Offset: 0

Views

Author

Peter Bala, Jul 21 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355797 (row 4).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 3: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
O.g.f.: A(x) = ( Sum_{k >= 0} t(k+3)/t(3)*x^k )/( Sum_{k >= 0} t(k+2)/t(2)*x^k ).
A(x)/(1 - 8*x*A(x)) = Sum_{k >= 0} t(k+3)/t(3)*x^k.
A(x) = 1/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - 17*x/(1 + 17*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 8*x*R(n,x)^2 - (1 + 5*x)*R(n,x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 11*x/(1 - 6*x/(1 - 14*x/(1 - 9*x/(1 - 17*x/(1 - 12*x/(1 - ...)))))))), a continued fraction of Stieltjes type.

A355797 Row 4 of A355793.

Original entry on oeis.org

1, 3, 51, 1119, 29103, 859143, 28091463, 1002057591, 38606468343, 1595167432599, 70315835952471, 3293268346004439, 163337193581191575, 8554718468806548951, 471976737725208306327, 27369722655919760451159, 1664858070989667129693975, 106029602841882346657155543
Offset: 0

Views

Author

Peter Bala, Jul 21 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355796 (row 3).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 4: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
O.g.f.: A(x) = ( Sum_{k >= 0} t(k+4)/t(4)*x^k )/( Sum_{k >= 0} t(k+3)/t(3)*x^k ).
A(x)/(1 - 11*x*A(x)) = Sum_{k >= 0} t(k+4)/t(4)*x^k.
A(x) = 1/(1 + 11*x - 14*x/(1 + 14*x -17*x/(1 + 17*x - 20*x/(1 + 20*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 11*x*A(x)^2 - (1 + 8*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 14*x/(1 - 6*x/(1 - 17*x/(1 - 9*x/(1 - 20*x/(1 - 12*x/(1 - 23*x/(1 - ...))))))))), a continued fraction of Stieltjes type.
Showing 1-9 of 9 results.

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