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.

A063265 Septinomial (also called heptanomial) coefficient array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 33, 36, 37, 36, 33, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 116, 149, 180, 206, 224, 231, 224, 206, 180, 149, 116, 84, 56, 35
Offset: 0

Views

Author

Wolfdieter Lang, Jul 24 2001

Keywords

Comments

The sequence of step width of this staircase array is [1,6,6,...], hence the degree sequence for the row polynomials is [0,6,12,18,...]= A008588.
The column sequences (without leading zeros) are for k=0..6 those of the lower triangular array A007318 (Pascal) and for k=7..9: A063267, A063417, A063418. Row sums give A000420 (powers of 7). Central coefficients give A025012.

Examples

			Triangle begins:
  {1};
  {1, 1, 1, 1, 1, 1, 1};
  {1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1};
  ...
N7(k,x)= 1 for k=0..6, N7(7,x)= 6-15*x+20*x^2-15*x^3+6*x^4-x^5 (from A063266).

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.

Links

Crossrefs

The q-nomial arrays are for q=2..8: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890.

Programs

  • Maple
    #Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 7-nomials as a table
r := 7:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013
  • Mathematica
    Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)

Formula

a(n, k)=0 if n=-1 or k<0 or k >= 6*n; a(0, 0)=1; a(n, k)= sum(a(n-1, k-j), j=0..6) else.
G.f. for row n: (sum(x^j, j=0..6))^n.
G.f. for column k: (x^(ceiling(k/6)))*N7(k, x)/(1-x)^(k+1) with the row polynomials of the staircase array A063266(k, m).
T(n,k) = Sum_{i = 0..floor(k/7)} (-1)^i*binomial(n,i)*binomial(n+k-1-7*i,n-1) for n >= 0 and 0 <= k <= 6*n. - Peter Bala, Sep 07 2013

A063417 Ninth column (k=8) of septinomial array A063265.

Original entry on oeis.org

5, 36, 149, 470, 1251, 2954, 6371, 12789, 24210, 43637, 75438, 125801, 203294, 319545, 490058, 735182, 1081251, 1561914, 2219675, 3107664, 4291661, 5852396, 7888149, 10517675, 13883480, 18155475, 23535036
Offset: 0

Views

Author

Wolfdieter Lang, Jul 24 2001

Keywords

Links

Crossrefs

Cf. A063267.

Programs

  • Mathematica
    Table[Total[Table[Binomial[n+2,i],{i,2,8}]{5,21,35,35,21,7,1}],{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{5,36,149,470,1251,2954,6371,12789,24210},30] (* Harvey P. Dale, Aug 22 2012 *)

Formula

a(n) = A063265(n+2,8) = (n+1)*(n+2)*(n^6 +41*n^5 +701*n^4 +6439*n^3 +33930*n^2 +100008*n +100800)/8!.
G.f.: (5-9*x+5*x^2+5*x^3-9*x^4+5*x^5-x^6)/(1-x)^9; the numerator polynomial is N6(8,x) from row n=8 of array A063266.
a(n) = 5*C(n+2,2) + 21*C(n+2,3) + 35*C(n+2,4) + 35*C(n+2,5) + 21*C(n+2,6) + 7*C(n+2,7) + C(n+2,8) (see comment in A213889). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(0)=5, a(1)=36, a(2)=149, a(3)=470, a(4)=1251, a(5)=2954, a(6)=6371, a(7)=12789, a(8)=24210, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)- 126*a(n-4)+ 126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Aug 22 2012

A063267 Eighth column (k=7) of septinomial array A063265.

Original entry on oeis.org

6, 33, 116, 325, 786, 1709, 3424, 6426, 11430, 19437, 31812, 50375, 77506, 116265, 170528, 245140, 346086, 480681, 657780, 888009, 1184018, 1560757, 2035776, 2629550, 3365830, 4272021, 5379588, 6724491, 8347650
Offset: 0

Views

Author

Wolfdieter Lang, Jul 24 2001

Keywords

Links

Crossrefs

Cf. A000579 (column k=6 of A063265).

Programs

  • Maple
    [seq((binomial(n+7,n)-binomial(n+1,n)),n=1..29)]; # Zerinvary Lajos, Jun 23 2006
  • Mathematica
    Table[Binomial[n+7,n]-Binomial[n+1,n],{n,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{6,33,116,325,786,1709,3424,6426},30] (* Harvey P. Dale, Jan 06 2012 *)

Formula

a(n)= A063265(n+2, 7)= (n+1)*(n+2)*(n+10)*(n^4 + 22*n^3 + 193*n^2 + 792*n + 1512)/7!.
G.f.: (2-x)*(1-x+x^2)*(3-3*x+x^2)/(1-x)^8; the numerator polynomial is N7(7, x) = 6 - 15*x + 20*x^2 - 15*x^3 + 6*x^4 - x^5 from row n=7 of array A063266.
a(n) = binomial(n+7,n) - binomial(n+1,n). - Zerinvary Lajos, Jun 23 2006
a(n) = binomial(n+7,n) + binomial(n+6,n) + binomial(n+5,n) + binomial(n+4,n) + binomial(n+3,n) + binomial(n+2,n). - Zerinvary Lajos, Jun 23 2006
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8); a(0)=6, a(1)=33, a(2)=116, a(3)=325, a(4)=786, a(5)=1709, a(6)=3424, a(7)=6426. - Harvey P. Dale, Jan 06 2012

Extensions

More terms from Zerinvary Lajos, Jun 23 2006

A063418 Tenth column (k=9) of septinomial array A063265.

Original entry on oeis.org

4, 37, 180, 640, 1876, 4809, 11152, 23905, 48070, 91652, 167024, 292747, 495950, 815390, 1305328, 2040374, 3121472, 4683215, 6902700, 10010154, 14301584, 20153727, 28041600, 38558975, 52442130
Offset: 0

Views

Author

Wolfdieter Lang, Jul 24 2001

Keywords

Crossrefs

Cf. A063417, A213889.

Formula

a(n) = A063265(n+2, 9) = (n+1)*(n+2)*(n+3)*(n^6+48*n^5+967*n^4+10548*n^3+66676*n^2+239280*n+241920)/9!.
G.f.: (4-3*x-10*x^2+25*x^3-24*x^4+11*x^5-2*x^6)/(1-x)^10; the numerator polynomial is N7(8, x) from row n=8 of array A063266.
a(n) = 4*C(n+2,2) + 25*C(n+2,3) + 56*C(n+2,4) + 70*C(n+2,5) + 56*C(n+2,6) + 28*C(n+2,7) + 8*C(n+2,8) + C(n+2,9) (see comment in A213889). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
Showing 1-4 of 4 results.

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