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).

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).

A117338 Row sums of triangle A117335.

Original entry on oeis.org

1, 2, 1, 4, -9, 62, -399, 3048, -26045, 247298, -2582027, 29416876, -363327081, 4837734374, -69105690039, 1054490587824, -17122237729589, 294828907099274, -5366869867749347, 102988994579465716, -2078107926978317889, 43988545301378533742
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2006

Keywords

Examples

			A(x) = 1 + 2*x + x^2 + 4*x^3 - 9*x^4 + 62*x^5 - 399*x^6 +-...
= 1/(1-x)/(1 - x + 2*x^2 - 6*x^3 + 24*x^4 - 120*x^5 +-...)

Crossrefs

Cf. A117335 (triangle), A117336 (column 1), A117337 (column 2).

Programs

  • PARI
    a(n)=polcoeff(1/((1-x)*sum(k=0,n,(-1)^k*k!*x^k)+x*O(x^n)),n)

Formula

G.f.: A(x) = 1/(1-x)/[Sum_{n>=0} (-1)^n*n!*x^n].

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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