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 1-4 of 4 results.

A117335 Matrix inverse of triangle A117334.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 6, -4, 1, 1, -27, 24, -8, 1, 1, 164, -157, 66, -13, 1, 1, -1133, 1176, -571, 146, -19, 1, 1, 8930, -9853, 5335, -1621, 281, -26, 1, 1, -78739, 91498, -53989, 18635, -3909, 491, -34, 1, 1, 768276, -933451, 591157, -225490, 54430, -8382, 799, -43, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2006

Keywords

Examples

			Triangle begins:
1;
1,1;
1,-1,1;
1,6,-4,1;
1,-27,24,-8,1;
1,164,-157,66,-13,1;
1,-1133,1176,-571,146,-19,1;
1,8930,-9853,5335,-1621,281,-26,1;
1,-78739,91498,-53989,18635,-3909,491,-34,1;
1,768276,-933451,591157,-225490,54430,-8382,799,-43,1; ...
Matrix inverse yields A117334:
1;
-1,1;
-2,1,1;
-3,-2,4,1;
-4,-13,8,8,1;
-5,-44,-3,38,13,1;
-6,-123,-117,125,101,19,1;
-7,-314,-718,205,594,213,26,1; ...
in which column k+1 is the Binomial transform of column k
preceded by a zero (includes the k zeros above diagonal):
column 1 = BINOMIAL[0, 1,-1,-2,-3,-4,-5,...]
= [0,1,1,-2,-13,-44,-123,-314,-761,...];
column 2 = BINOMIAL[0, 0,1,1,-2,-13,-44,-123,-314,...]
= [0,0,1,4,8,-3,-117,-718,-3314,...].

Crossrefs

Cf. A117334 (inverse), A117336 (column 1), A117337 (column 2), A117338 (row sums).

A117336 Column 1 of triangle A117335.

Original entry on oeis.org

0, 1, -1, 6, -27, 164, -1133, 8930, -78739, 768276, -8221025, 95775678, -1207356591, 16381797128, -238127430413, 3693030808322, -60880099651291, 1063264668030924, -19614400226382041, 381146295159812646, -7782393142569110535, 166592263396576831760
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2006

Keywords

Crossrefs

Cf. A104981, A117335 (triangle), A117337 (column 2), A117338 (row sums).

Programs

  • PARI
    {a(n)=if(n<1,0,(vector(n+1,k,(-1)^k)*matrix(n+1,n+1,m,j,if(m==j,1,if(m==j+1,-m+1, -polcoeff((1-1/sum(i=0,m,i!*x^i))/x+O(x^m),m-j-1))))^-1)[2])}

Formula

a(n) = Sum_{k=0..n} (-1)^(k+1)*A104981(k).

A117337 Column 2 of triangle A117335.

Original entry on oeis.org

0, 0, 1, -4, 24, -157, 1176, -9853, 91498, -933451, 10386908, -125292129, 1629597910, -22746009423, 339288465552, -5388025686685, 90782990910394, -1617895300046011, 30412104692215604, -601422325704360009, 12483373271662252222, -271375594220745999111
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2006

Keywords

Crossrefs

Cf. A117335 (triangle), A117336 (column 1), A117338 (row sums).

A117334 Triangle, read by rows, where column 0 is [1,-1,-2,-3,...,-n,...] and column k+1 is generated by the binomial transform of column k preceded by a zero (column k includes the k zeros above the main diagonal).

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -3, -2, 4, 1, -4, -13, 8, 8, 1, -5, -44, -3, 38, 13, 1, -6, -123, -117, 125, 101, 19, 1, -7, -314, -718, 205, 594, 213, 26, 1, -8, -761, -3314, -954, 2787, 1822, 393, 34, 1, -9, -1784, -13481, -12644, 9717, 12987, 4507, 663, 43, 1, -10, -4087, -51055, -90625, 12247, 79419, 43282, 9727, 1048
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2006

Keywords

Comments

Row sums are all zeros after row 0.

Examples

			To generate, start with [1,-1,-2,-3,-4,...] in column 0.
Then column 1 = BINOMIAL[0, 1,-1,-2,-3,-4,-5,...]
= [0,1,1,-2,-13,-44,-123,-314,-761,...];
column 2 = BINOMIAL[0, 0,1,1,-2,-13,-44,-123,-314,...]
= [0,0,1,4,8,-3,-117,-718,-3314,-13481,...];
column 3 = BINOMIAL[0, 0,0,1,4,8,-3,-117,-718,...]
= [0,0,0,1,8,38,125,205,-954,-12644,-90625,...]; etc.
Triangle begins:
1;
-1,1;
-2,1,1;
-3,-2,4,1;
-4,-13,8,8,1;
-5,-44,-3,38,13,1;
-6,-123,-117,125,101,19,1;
-7,-314,-718,205,594,213,26,1;
-8,-761,-3314,-954,2787,1822,393,34,1;
-9,-1784,-13481,-12644,9717,12987,4507,663,43,1;
-10,-4087,-51055,-90625,12247,79419,43282,9727,1048,53,1; ...

Crossrefs

Cf. A117335 (matrix inverse), A117336, A117337, A117338.

Programs

  • PARI
    T(n,k)=if(n

Formula

T(n,k) = Sum_{i=k..n} C(n,i)*T(i-1,k-1) for k>0, with T(0,0)=1 and T(n,0)=-n for n>0.
Showing 1-4 of 4 results.

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

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