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.

A086672 Stirling1 transform of Catalan numbers: Sum_{k=0..n} Stirling1(n,k)*binomial(2*k,k)/(k+1).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, -5, 29, -196, 1518, -13266, 129163, -1386572, 16270671, -207195495, 2845705719, -41930575740, 659781404944, -11041824881696, 195839234324062, -3669384701403344, 72423881548363354, -1501924519315744146, 32649768696532126439, -742432111781693213350
Offset: 0

Views

Author

Vladeta Jovovic, Sep 12 2003

Keywords

Comments

1, 1, 1, 0, 1, -5, 29, -196, ... is the Stirling1 transform of the Motzkin numbers A001006. - Philippe Deléham, May 27 2015

Links

Crossrefs

Cf. A000108, A001006, A008275, A048994, A064856.
Cf. A201950, A201952, A306335.

Programs

  • Mathematica
    Table[Sum[StirlingS1[n, k] * CatalanNumber[k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Aug 04 2021 *)
  • PARI
    a(n)={sum(k=0, n, stirling(n,k,1) * binomial(2*k, k) / (k+1))} \\ Andrew Howroyd, Jan 27 2020

Formula

E.g.f.: hypergeom([1/2], [2], 4*log(1+x)) = (1+x)^2*(BesselI(0, 2*log(1+x))-BesselI(1, 2*log(1+x))).
Let C(m) be the m-th Catalan number, A000108(m). Let S(m, n) = an unsigned Stirling number of the first kind. Then a(m) = sum{k=0 to m} S(m, k) C(k) (-1)^(k+m). - Leroy Quet, Jan 23 2004
E.g.f. f(x) satisfies f(x) = 1 + integral{0 to x} f(y) f((x-y)/(1+y))/(1+y) dy. - Leroy Quet, Jan 25 2004
a(n) = Sum_{k = 0..n} A048994(n, k) * A000108(k). - Philippe Deléham, May 27 2015
a(n+1) = Sum_{k = 0..n} A048994(n,k) * A001006(k). - Philippe Deléham, May 27 2015
For n > 1, a(n) = (A201950(n+1) - (3*n-2)*A201950(n) + n*(3*n-7)*A201950(n-1) - (n-4)*(n-1)*n*A201950(n-2)) * (-1)^n/2. - Vaclav Kotesovec, May 04 2024

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 27 2020

A201949 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-1} (1 + k*y + y^2) for n>0 with a single '1' in row 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 3, 5, 6, 5, 3, 1, 1, 6, 15, 24, 28, 24, 15, 6, 1, 1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1, 1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1, 1, 21, 182, 861, 2520, 5019, 7377, 8358, 7377, 5019, 2520, 861, 182, 21, 1, 1, 28, 330, 2156, 8729, 23520, 45030, 65016, 73260, 65016
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

The formula for the main diagonal, BesselI(0, 2*log(1 - x)), was found by Ilya Gutkovskiy (see A201950). - Paul D. Hanna, Feb 24 2019

Examples

			E.g.f.: A(x,y) = 1 + (1 + y^2)*x + (1 + y + 2*y^2 + y^3 + y^4)*x^2/2! + (1 + 3*y + 5*y^2 + 6*y^3 + 5*y^4 + 3*y^5 + y^6)*x^3/3! + (1 + 6*y + 15*y^2 + 24*y^3 + 28*y^4 + 24*y^5 + 15*y^6 + 6*y^7 + y^8)*x^4/4! + (1 + 10*y + 40*y^2 + 90*y^3 + 139*y^4 + 160*y^5 + 139*y^6 + 90*y^7 + 40*y^8 + 10*y^9 + y^10)*x^5/5! + ...
which equals the power series expansion in x of the series given by
A(x,y)  =  Sum_{n>=0} log(1 - x*y)^(2*n) / (n!^2)  -  (1/y + y) * Sum_{n>=0} log(1 - x*y)^(2*n+1) / (n!*(n+1)!)  +  (1/y^2 + y^2) * Sum_{n>=0} log(1 - x*y)^(2*n+2) / (n!*(n+2)!)  -  (1/y3 + y^3) * Sum_{n>=0} (-log(1 - x*y))^(2*n+3) / (n!*(n+3)!)  +  (1/y^4 + y^4) * Sum_{n>=0} log(1 - x*y)^(2*n+4) / (n!*(n+4)!) + ...
Triangle begins:
[1],
[1, 0, 1],
[1, 1, 2, 1, 1],
[1, 3, 5, 6, 5, 3, 1],
[1, 6, 15, 24, 28, 24, 15, 6, 1],
[1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1],
[1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1],
[1, 21, 182, 861, 2520, 5019, 7377, 8358, 7377, 5019, 2520, 861, 182, 21, 1],
[1, 28, 330, 2156, 8729, 23520, 45030, 65016, 73260, 65016, 45030, 23520, 8729, 2156, 330, 28, 1], ...
such that the g.f. of row n equals Product_{k=0..n-1} (1 + k*x + x^2) for n>0.
RELATED SERIES.
The e.g.f. may be defined by A(x,y) = x / Series_Reversion( F(x,y) )
where F(x,y) is the e.g.f. of triangle A324305 and equals
F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)
so that
F(x,y) = x + (2*y^2 + 2)*x^2/2! + (9*y^4 + 3*y^3 + 18*y^2 + 3*y + 9)*x^3/3! + (64*y^6 + 48*y^5 + 200*y^4 + 96*y^3 + 200*y^2 + 48*y + 64)*x^4/4! + (625*y^8 + 750*y^7 + 2775*y^6 + 2280*y^5 + 4300*y^4 + 2280*y^3 + 2775*y^2 + 750*y + 625)*x^5/5! + ...
where F(x,y) = Series_Reversion( x/A(x,y) ).
RELATED TRIANGLE.
Triangle A324305 of coefficients in F(x,y) such that F(x/A(x,y),y) = x begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776; ...
where the g.f. of row n is Product_{k=0..n-2} (n + k*y + n*y^2) for n >= 1.

Links

Crossrefs

Cf. A201950, A201951; diagonals: A201952, A201953.
Cf. A324305.

Programs

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

Formula

Row sums yield the factorials.
Central terms in rows form A201950.
Antidiagonal sums yield A201951.
GENERATING FUNCTIONS.
E.g.f.: A(x,y) = 1/(1 - x*y)^(1/y + y). - Paul D. Hanna, Mar 02 2019
E.g.f.: A(x,y) = Sum_{k>=0} (1/y^k + y^k)/2^(0^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!). - Paul D. Hanna, Feb 24 2019
E.g.f.: A(x,y) = x / Series_Reversion( F(x,y) ) such that F(x/A(x,y),y) = x, where F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) is the e.g.f. of A324305. - Paul D. Hanna, Feb 28 2019
E.g.f. of diagonal k: (1/y^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!) for k >= 0. - Paul D. Hanna, Feb 24 2019
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = exp(x).
E.g.f. at y = 1: A(x,y=1) = 1/(1-x)^2.
E.g.f. at y = 2: A(x,y=2) = 1/(1-2*x)^(5/2).

A201951 G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + k*x + x^2).

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 33, 85, 234, 675, 2032, 6367, 20677, 69442, 240529, 857634, 3141970, 11808611, 45464065, 179088744, 720947705, 2962994169, 12420658682, 53061133078, 230828047288, 1021809688593, 4599749893986, 21043392417004, 97784119963565, 461277854065112
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

Equals the antidiagonal sums of irregular triangle A201949.

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 6*x^4 + 13*x^5 + 33*x^6 + 85*x^7 +...
where the g.f. equals the series:
A(x) = 1 + x*(1+x^2) + x^2*(1+x^2)*(1+x+x^2) + x^3*(1+x^2)*(1+x+x^2)*(1+2*x+x^2) + x^4*(1+x^2)*(1+x+x^2)*(1+2*x+x^2)*(1+3*x+x^2) +...

Links

Crossrefs

Cf. A201949, A201950.

Programs

  • PARI
    {a(n)=sum(k=0,n,polcoeff(prod(j=0,n-k-1,1+j*x+x^2),k))}
  • PARI
    {a(n)=polcoeff(sum(m=0,n,x^m*prod(j=0,m-1,1+j*x+x^2))+x*O(x^n),n)}
  • PARI
    {a(n)=local(CF=x+x*O(x)); for(k=1, n, CF=x*(1+(n-k)*x+x^2)/(1+x*(1+(n-k)*x+x^2)-CF)); polcoeff(1/(1-CF), n, x)}

Formula

G.f.: A(x) = 1/(1 - x*(1+x^2)/(1+x*(1+x^2) - x*(1+x+x^2)/(1+x*(1+x+x^2) - x*(1+2*x+x^2)/(1+x*(1+2*x+x^2) - x*(1+3*x+x^2)/(1+x*(1+3*x+x^2) +...))))), a continued fraction.
G.f.: A(x) =1 + x*(1+x^2)/(G(0) - x*(1+x^2)) ; G(k)= k*x^2 + 1 + x + x^3 - x*(1+x+x^2+x*k)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 28 2011

A201952 A diagonal of irregular triangle A201949.

Original entry on oeis.org

1, 1, 5, 24, 139, 945, 7377, 65016, 638418, 6910650, 81747665, 1049089470, 14516096009, 215419836359, 3412889885571, 57492203734320, 1026121982213480, 19342642266760680, 383995631680561234, 8007915240045479980, 175020604366224762038, 4000551483475536398178
Offset: 1

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

G.f. of row n in triangle A201949 equals Product_{k=0..n-1} (1 + k*x + x^2).

Examples

			E.g.f.: A(x) = x + x^2/2! + 5*x^3/3! + 24*x^4/4! + 139*x^5/5! + 945*x^6/6! + 7377*x^7/7! + 65016*x^8/8! + 638418*x^9/9! + 6910650*x^10/10! + ...
Triangle A201949 begins:
[1],
[(1), 0, 1],
[1,(1), 2, 1, 1],
[1, 3, (5), 6, 5, 3, 1],
[1, 6, 15, (24), 28, 24, 15, 6, 1],
[1, 10, 40, 90,(139), 160, 139, 90, 40, 10, 1], ...
where coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A201949, A201950, A201951, A201953.

Programs

  • PARI
    {a(n) = polcoeff( prod(j=0, n-1, 1 + j*x + x^2), n-1)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} -log(1 - x)^(2*n+1) / (n!*(n+1)!). - Paul D. Hanna, Feb 25 2019
a(n) = [x^(n-1)] Product_{k=0..n-1} (1 + k*x + x^2).
a(n) = (n-1)*a(n-1) + A201950(n-1) + A201953(n-1).

Extensions

Offset changed to 1 to agree with the e.g.f. - Paul D. Hanna, Feb 25 2019

A201826 Central coefficients in Product_{k=1..n} (1 + k*y + y^2).

Original entry on oeis.org

1, 1, 4, 18, 100, 660, 5038, 43624, 422252, 4516380, 52885644, 672781824, 9238314358, 136175455234, 2144494356834, 35930786795040, 638168940129732, 11976278012219556, 236791150694618872, 4919643784275283480, 107152493449339765396, 2441410548192907949196
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 100*x^4/4! + 660*x^5/5! + 5038*x^6/6! + 43624*x^7/7! + 422252*x^8/8! + 4516380*x^9/9! + 52885644*x^10/10! + ...
The coefficients in Product_{k=1..n} (1 + k*y + y^2), n>=0, form triangle A249790:
[1];
[1, 1, 1];
[1, 3, 4, 3, 1];
[1, 6, 14, 18, 14, 6, 1];
[1, 10, 39, 80, 100, 80, 39, 10, 1];
[1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1];
[1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1];
[1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
in which the central terms of the rows form this sequence.

Crossrefs

Cf. A249790, A202474, A201950, A202476.

Programs

  • Mathematica
    Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,1,n}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)
  • PARI
    {a(n) = polcoeff(prod(k=1,n, 1 + k*x + x^2 +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = n!*polcoeff( sum(m=0, n, log(1 - x +x*O(x^n))^(2*m)/m!^2 ) / (1 - x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 02 2019

Formula

E.g.f.: 1/(1-x) * Sum_{n>=0} log(1 - x)^(2*n) / n!^2. - Paul D. Hanna, Mar 02 2019

A201953 A diagonal of irregular triangle A201949.

Original entry on oeis.org

1, 3, 15, 90, 629, 5019, 45030, 448776, 4919321, 58825415, 762089899, 10633219662, 158974192987, 2535484008225, 42970371055268, 771162539117408, 14609924404202130, 291386317037291622, 6102681801481066642, 133910606028043519500, 3072216586896101950757
Offset: 2

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

G.f. of row n in triangle A201949 equals Product_{k=0..n-1} (1 + k*x + x^2).

Examples

			E.g.f.: A(x) = x^2/2! + 3*x^3/3! + 15*x^4/4! + 90*x^5/5! + 629*x^6/6! + 5019*x^7/7! + 45030*x^8/8! + 448776*x^9/9! + 4919321*x^10/10! + ...
Triangle A201949 begins:
[1],
[1, 0, 1],
[(1), 1, 2, 1, 1],
[1,(3), 5, 6, 5, 3, 1],
[1, 6, (15), 24, 28, 24, 15, 6, 1],
[1, 10, 40, (90), 139, 160, 139, 90, 40, 10, 1],
[1, 15, 91, 300, (629), 945, 1078, 945, 629, 300, 91, 15, 1],  ...
where coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A201949, A201950, A201951, A201952.

Programs

  • PARI
    {a(n) = polcoeff( prod(j=0, n-1, 1 + j*x + x^2), n-2)}
for(n=2,30,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} log(1 - x)^(2*n+2) / (n!*(n+2)!). - Paul D. Hanna, Feb 25 2019
a(n) = [x^(n-2)] Product_{k=0..n-1} (1 + k*x + x^2).

Extensions

Offset changed to 2 to agree with the e.g.f. - Paul D. Hanna, Feb 25 2019

A202474 Coefficients of y^(n-1) in Product_{k=1..n} (1 + k*y + y^2) for n >= 1.

Original entry on oeis.org

1, 3, 14, 80, 539, 4179, 36630, 358056, 3860922, 45519870, 582466235, 8038684290, 119018991779, 1881685721265, 31638175704546, 563703015007056, 10609073237333432, 210305960538762456, 4379808881917047898, 95604092878386437940, 2182706554812339958778
Offset: 1

Views

Author

Paul D. Hanna, Dec 19 2011

Keywords

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 14*x^3/3! + 80*x^4/4! + 539*x^5/5! + 4179*x^6/6! + 36630*x^7/7! + 358056*x^8/8! + 3860922*x^9/9! + 45519870*x^10/10! + ...
The coefficients in Product_{k=1..n} (1+k*x+x^2), n>=0, form the triangle:
[1];
[(1), 1, 1];
[1,(3), 4, 3, 1];
[1, 6, (14), 18, 14, 6, 1];
[1, 10, 39, (80), 100, 80, 39, 10, 1];
[1, 15, 90, 285, (539), 660, 539, 285, 90, 15, 1];
[1, 21, 181, 840, 2339, (4179), 5038, 4179, 2339, 840, 181, 21, 1];
[1, 28, 329, 2128, 8400, 21392, (36630), 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
the coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A201826, A201950, A202476.

Programs

  • Mathematica
    Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,1,n+1}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)
  • PARI
    {a(n) = polcoeff(prod(k=1,n,1 + k*x + x^2 +x*O(x^n)),n-1)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f.: 1/(1-x) * Sum_{n>=0} log(1 - x)^(2*n+1) / (n!*(n+1)!).

Extensions

Changed offset to 1 to agree with e.g.f. - Paul D. Hanna, Mar 02 2019

A324304 a(n) = [y^(n-1)] Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a(1) = 1.

Original entry on oeis.org

1, 0, 18, 96, 4300, 81360, 3604342, 128389632, 6704335980, 346778956800, 21896347260084, 1459386186255360, 110117675704707190, 8898156449299703040, 786739773441598071750, 74406732202318884372480, 7565016269351818379826372, 818338704493281924572946432, 94154670956813022045927404464, 11458715042302170139584184320000, 1472412964588453156024745207931636
Offset: 1

Views

Author

Paul D. Hanna, Feb 28 2019

Keywords

Examples

			E.g.f.: A(x) = x + 18*x^3/3! + 96*x^4/4! + 4300*x^5/5! + 81360*x^6/6! + 3604342*x^7/7! + 128389632*x^8/8! + 6704335980*x^9/9! + 346778956800*x^10/10! + 21896347260084*x^11/11! + 1459386186255360*x^12/12! + ...
RELATED TRIANGLE.
Triangle A324305 of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649; ...
in which the central terms, A324305(n, n-1) for n >= 1, form this sequence.

Links

Crossrefs

Cf. A324305.
Cf. A201950 (variant).

Programs

  • Mathematica
    Flatten[{1, Table[Coefficient[Expand[Product[(n + k*y + n*y^2), {k, 0, n-2}]], y^(n-1)], {n, 2, 20}]}] (* Vaclav Kotesovec, Mar 13 2019 *)
  • PARI
    {A324305(n, k) = polcoeff( prod(j=0, n-2,  n + j*y + n*y^2), k, y)}
{a(n) = A324305(n, n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

a(n) = A324305(n, n-1) for n >= 1.
a(n) ~ c * n! * (27/4)^n / n^2, where c = 1/(6*Pi*sqrt(3*log(3/2))) = 0.04810181967270783985882272373499905248047631331... - Vaclav Kotesovec, Mar 13 2019, updated Mar 17 2024
Showing 1-8 of 8 results.

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

Content is available under The OEIS End-User License Agreement.