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 1-3 of 3 results.

A179748 Triangle T(n,k) read by rows. T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 5, 4, 1, 1, 1, 2, 6, 9, 5, 1, 1, 1, 2, 6, 15, 14, 6, 1, 1, 1, 2, 6, 20, 29, 20, 7, 1, 1, 1, 2, 6, 23, 49, 49, 27, 8, 1, 1, 1, 2, 6, 24, 71, 98, 76, 35, 9, 1, 1, 1, 2, 6, 24, 91, 169, 174, 111, 44, 10, 1, 1, 1, 2, 6, 24, 106, 259, 343, 285, 155, 54, 11, 1
Offset: 1

Views

Author

Mats Granvik, Jul 26 2010

Keywords

Comments

Recurrence is half of the recurrence for divisibility in A051731. That is, without subtracting (Sum_{i=1..k-1} T(n-i,k)).
Rows tend to factorial numbers.
Row sums are A177510.

Examples

			Triangle begins:
01: 1;
02: 1, 1;
03: 1, 1, 1;
04: 1, 1, 2, 1;
05: 1, 1, 2, 3,  1;
06: 1, 1, 2, 5,  4,   1;
07: 1, 1, 2, 6,  9,   5,   1;
08: 1, 1, 2, 6, 15,  14,   6,    1;
09: 1, 1, 2, 6, 20,  29,  20,    7,    1;
10: 1, 1, 2, 6, 23,  49,  49,   27,    8,    1;
11: 1, 1, 2, 6, 24,  71,  98,   76,   35,    9,    1;
12: 1, 1, 2, 6, 24,  91, 169,  174,  111,   44,   10,    1;
13: 1, 1, 2, 6, 24, 106, 259,  343,  285,  155,   54,   11,    1;
14: 1, 1, 2, 6, 24, 115, 360,  602,  628,  440,  209,   65,   12,   1;
15: 1, 1, 2, 6, 24, 119, 461,  961, 1230, 1068,  649,  274,   77,  13,   1;
16: 1, 1, 2, 6, 24, 120, 551, 1416, 2191, 2298, 1717,  923,  351,  90,  14,  1;
17: 1, 1, 2, 6, 24, 120, 622, 1947, 3606, 4489, 4015, 2640, 1274, 441, 104, 15, 1;
...

Crossrefs

Cf. A175105, A051731, A179749, A179750, A000142.

Programs

  • Sage
    @CachedFunction
def T(n, k): # A179748
    if n == 0:  return int(k==0);
    if k == 1:  return int(n>=1);
    return sum( T(n-i, k-1) for i in [1..k-1] );
for n in [1..15]: print([ T(n, k) for k in [1..n] ])
# Joerg Arndt, Mar 24 2014

Formula

T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).

A127926 G.f.: 1-q = Sum_{k>=0} a(k)*q^k*Faq(k+1,q), where Faq(n,q) is the q-factorial of n.

Original entry on oeis.org

1, -1, 1, -2, 4, -7, 11, -18, 35, -76, 166, -358, 775, -1686, 3638, -7716, 16108, -33349, 69022, -143605, 301179, -636932, 1355855, -2896168, 6186750, -13183426, 27988755, -59197443, 124824911, -262699256, 552438175, -1162010894, 2446434685, -5156873960
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2007

Keywords

Examples

			Define Faq(n,q) = Product_{i=1..n} (1-q^i)/(1-q) for n>0, Faq(0,q)=1.
Then coefficients of q in a(k)*q^k*Faq(k+1,q) begin as follows:
k=0: 1;
k=1: .. -1, -1;
k=2: ....... 1, 2, 2,. 1;
k=3: ......... -2,-6,-10,-12,-10,. -6,. -2;
k=4: ............. 4, 16, 36, 60,. 80,. 88,.. 80, ...;
k=5: ................ -7,-35,-98,-203,-343, -497, ...;
k=6: .................... 11, 66, 220, 539, 1078, ...;
k=7: ....................... -18,-126,-486,-1368, ...;
k=8: ............................. 35, 280, 1225, ...;
k=9: ................................. -76, -684, ...;
k=10: ...................................... 166, ...;
Sums cancel down column j for j>1, leaving 1-q.

Links

Crossrefs

First column of A179750. [From Mats Granvik, Jul 26 2010]
Cf. A129273.

Programs

  • PARI
    {a(n)=if(n==0,1,polcoeff(1-q- sum(k=0,n-1,a(k)*q^k*prod(j=1,k+1,(1-q^j)/(1-q+q*O(q^(n-k))))),n,q))}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: 1-q = Sum_{k>=0} a(k)*q^k*Product_{i=1..k+1} (1-q^i)/(1-q).

A179749 Subsequence of A179748. Equals A179748 beginning at the second column.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 5, 4, 1, 1, 2, 6, 9, 5, 1, 1, 2, 6, 15, 14, 6, 1, 1, 2, 6, 20, 29, 20, 7, 1, 1, 2, 6, 23, 49, 49, 27, 8, 1, 1, 2, 6, 24, 71, 98, 76, 35, 9, 1, 1, 2, 6, 24, 91, 169, 174, 111, 44, 10, 1, 1, 2, 6, 24, 106, 259, 343, 285, 155, 54, 11, 1, 1, 2, 6, 24, 115, 360
Offset: 1

Views

Author

Mats Granvik, Jul 26 2010

Keywords

Comments

Matrix inverse is A179750.

Examples

			Table begins:
1,
1,1,
1,2,1,
1,2,3,1,
1,2,5,4,1,
1,2,6,9,5,1,
1,2,6,15,14,6,1,
1,2,6,20,29,20,7,1,
1,2,6,23,49,49,27,8,1,
1,2,6,24,71,98,76,35,9,1,
1,2,6,24,91,169,174,111,44,10,1,
1,2,6,24,106,259,343,285,155,54,11,1,
1,2,6,24,115,360,602,628,440,209,65,12,1,
1,2,6,24,119,461,961,1230,1068,649,274,77,13,1,

Crossrefs

Cf. A179748, A179750.
Showing 1-3 of 3 results.

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