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 1-6 of 6 results.

A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 5, 8, 0, 0, 6, 15, 17, 16, 0, 0, 24, 62, 68, 49, 32, 0, 0, 120, 322, 359, 243, 129, 64, 0, 0, 720, 2004, 2308, 1553, 756, 321, 128, 0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256, 0, 0, 40320, 119664
Offset: 0

Views

Author

Philippe Deléham, Oct 19 2005

Keywords

Comments

Let R(m,n,k), 0<=k<=n, the Riordan array (1, x*g(x)) where g(x) is g.f. of the m-fold factorials . Then Sum_{k, 0<=k<=n} = R(m,n,k) = Sum_{k, 0<=k<=n} T(n,k)*m^(n-k).
For m = -1, R(-1,n,k) is A026729(n,k).
For m = 0, R(0,n,k) is A097805(n,k).
For m = 1, R(1,n,k) is A084938(n,k).
For m = 2, R(2,n,k) is A111106(n,k).

Examples

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 4;
.0, 0, 2, 5, 8;
.0, 0, 6, 15, 17, 16;
.0, 0, 24, 62, 68, 49, 32;
.0, 0, 120, 322, 359, 243, 129, 64;
.0, 0, 720, 2004, 2308, 1553, 756, 321, 128;
.0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256;
.0, 0, 40320, 119664, 148232, 105048, 49840, 18066, 5756, 1793, 512;
....................................................................
At y=2: Sum_{k=0..n} 2^k*T(n,k) = A113327(n) where (1 + 2*x + 8*x^2 + 36*x^3 +...+ A113327(n)*x^n +..) = 1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 +..) ).
At y=3: Sum_{k=0..n} 3^k*T(n,k) = A113328(n) where (1 + 3*x + 18*x^2 + 117*x^3 +...+ A113328(n)*x^n +..) = 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 +..) ).
At y=4: Sum_{k=0..n} 4^k*T(n,k) = A113329(n) where (1 + 4*x + 32*x^2 + 272*x^3 +...+ A113329(n)*x^n +..) = 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 +..) ).

Links

Crossrefs

Cf. A000045, A026729, A051295, A084938, A097805, A111106, A112934.
Cf. m-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542.
Cf. A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

Programs

  • Mathematica
    T[n_, k_] := Module[{x = X + X*O[X]^n, y = Y + Y*O[Y]^k}, A = 1/(1 - x*y*Sum[x^j*Product[y + i, {i, 0, j - 1}], {j, 0, n}]); Coefficient[ Coefficient[A, X, n], Y, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, from PARI *)
  • PARI
    {T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); A=1/(1-x*y*sum(j=0,n,x^j*prod(i=0,j-1,y+i))); return(polcoeff(polcoeff(A,n,X),k,Y))} (Hanna)

Formula

Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n, k) = A000045(n+1), Fibonacci numbers.
Sum_{k, 0<=k<=n} T(n, k) = A051295(n).
Sum_{k, 0<=k<=n} 2^(n-k)*T(n, k) = A112934(n).
T(0, 0) = 1, T(n, n) = 2^(n-1).
G.f.: A(x, y) = 1/(1 - x*y*Sum_{j>=0} (y-1+j)!/(y-1)!*x^j ). - Paul D. Hanna, Oct 26 2005

A113326 Table T(n,k), n>=1 and k>=0, read by antidiagonals, related to A111146.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 8, 5, 1, 4, 18, 36, 15, 1, 5, 32, 117, 176, 54, 1, 6, 50, 272, 801, 928, 235, 1, 7, 72, 525, 2400, 5724, 5296, 1237, 1, 8, 98, 900, 5675, 21792, 42633, 33024, 7790, 1, 9, 128, 1421, 11520, 62650, 203008, 331911, 227776, 57581
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			Table begins:
1,1,2,5,15,54,235,1237,7790,57581, 489231, ...
1,2,8,36,176,928,5296,33024,227776,1757504, ...
1,3,18,117,801,5724,42633,331911,2717874,23620329, ...
1,4,32,272,2400,21792,203008,1940224,19065344,193410560, ...
1,5,50,525,5675,62650,703975,8042625,93454750,1106250125, ...
1,6,72,900,11520,149904,1976400,26363232,355648320, ...
1,7,98,1421,21021,315168,4774021,72945859,1123559906, ...
1,8,128,2112,35456,601984,10306048,177639936,3080264704, ...
1,9,162,2997,56295,1067742,20392803,391614669,7555447854, ...
1,10,200,4100,85200,1785600,37644400,797224000,16946456000,
...

Crossrefs

Cf. A111146, A051295 (row n=1), A113327 (row n=2), A113328 (row n=3), A113329 (row n=4), A113330 (row n=5), A113331 (row n=6).

Programs

  • PARI
    {T(n,k)=local(y=Y+Y*O(Y^k)); polcoeff(1/(1-n/(n-1)!*y*sum(j=0,k,(n-1+j)!*y^j)),k,Y)}

Formula

T(n, k) = Sum_{j=0..k} n^j*A111146(k, j).
G.f. for row n: Sum_{k>=0}T(n, k)*y^k = 1/(1-n/(n-1)!*y*Sum_{j>=0}(n-1+j)!*y^j), for n>=1.

A113327 a(n) = Sum_{k=0..n} 2^k*A111146(n,k).

Original entry on oeis.org

1, 2, 8, 36, 176, 928, 5296, 33024, 227776, 1757504, 15269888, 149327616, 1632715520, 19758502912, 261836047360, 3763432774656, 58208166178816, 962637398577152, 16934963591229440, 315578267054112768
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 2*x + 8*x^2 + 36*x^3 + 176*x^4 + 928*x^5 +..) =
1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 + 5!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

Programs

  • PARI
    {a(n)=local(y=2,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - 2*x*Sum_{k>=0} (k+1)!*x^k ).

A113328 a(n) = Sum_{k=0..n} 3^k*A111146(n,k).

Original entry on oeis.org

1, 3, 18, 117, 801, 5724, 42633, 331911, 2717874, 23620329, 220260789, 2228505372, 24681015981, 300506801715, 4017984855786, 58675338993069, 928673101727001, 15804592586240220, 287174716511520033, 5538727108037507535
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 3*x + 18*x^2 + 117*x^3 + 801*x^4 + 5724*x^5 +..)
= 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 + 6!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113327 (y=2), A113329 (y=4), A113330 (y=5), A113331 (y=6).

Programs

  • PARI
    {a(n)=local(y=3,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - (3/2)*x*Sum_{k>=0} (k+2)!*x^k ).

A113329 a(n) = Sum_{k=0..n} 4^k*A111146(n,k).

Original entry on oeis.org

1, 4, 32, 272, 2400, 21792, 203008, 1940224, 19065344, 193410560, 2038078464, 22490167296, 262429339648, 3271314362368, 43955391856640, 640254018879488, 10121874150653952, 173145693892509696, 3186234896556752896
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 4*x + 32*x^2 + 272*x^3 + 2400*x^4 + 21792*x^5 +..)
= 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 + 7!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113327 (y=2), A113328 (y=3), A113330 (y=5), A113331 (y=6).

Programs

  • PARI
    {a(n)=local(y=4,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - (2/3)*x*Sum_{k>=0} (k+3)!*x^k ).

A113331 a(n) = Sum_{k=0..n} 6^k*A111146(n,k).

Original entry on oeis.org

1, 6, 72, 900, 11520, 149904, 1976400, 26363232, 355648320, 4854292416, 67114780416, 941774874624, 13451571452160, 196362144456192, 2945496714485760, 45717104468689920, 740282299231703040
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 6*x + 72*x^2 + 900*x^3 + 11520*x^4 + 149904*x^5 +..)
= 1/(1 - 6/5!*x*(5! + 6!*x + 7!*x^2 + 8!*x^3 + 9!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5).

Programs

  • PARI
    {a(n)=local(y=6,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - (1/20)*x*Sum_{k>=0} (k+5)!*x^k ).
Showing 1-6 of 6 results.

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