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.

A107028 Row sums of number triangle A107027.

Original entry on oeis.org

1, 3, 5, 9, 19, 49, 155, 591, 2651, 13689, 80009, 522193, 3762273, 29629483, 252966249, 2325176147, 22874076091, 239615657133, 2661015200515, 31207545002577, 385191232715737, 4988767088734625, 67615869255979521
Offset: 0

Views

Author

Paul Barry, May 09 2005

Keywords

Formula

a(n)=sum{k=0..n, (n-k-1)C((n-k)*k, k)-(n-k-2)*sum{j=0..k, C((n-k)*k, j)}}

A107029 Diagonal sums of number triangle A107027.

Original entry on oeis.org

1, 1, 3, 3, 5, 7, 11, 19, 35, 69, 151, 339, 829, 2111, 5607, 15737, 45289, 136477, 423279, 1353671, 4471673, 15114479, 52605755, 187162155, 681817179, 2539492249, 9646349779, 37415172217, 147745308403, 594368656811
Offset: 0

Views

Author

Paul Barry, May 09 2005

Keywords

Formula

a(n)=sum{k=0..floor(n/2), (n-2k-1)C((n-2k)*k, k)-(n-2k-2)*sum{j=0..k, C((n-2k)*k, j)}}

A047089 Array T read by antidiagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and touches the line y=x/2 only at lattice points.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 4, 7, 4, 4, 1, 1, 5, 11, 11, 8, 5, 1, 1, 6, 16, 22, 19, 13, 6, 1, 1, 7, 22, 38, 41, 19, 19, 7, 1, 1, 8, 29, 60, 79, 60, 38, 26, 8, 1, 1, 9, 37, 89, 139, 139, 98, 64, 34, 9, 1, 1, 10, 46, 126, 228, 278, 237, 98, 98, 43, 10, 1, 1, 11, 56, 172
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Comments from Timothy Y. Chow (tchow(AT)alum.mit.edu), Nov 15 2006 on this sequence and A107027. "If you replace "the line y = x/2" with "the line y = x/(n-1)" in the definition of this sequence, then the formula for T(h,k) becomes (h+k choose k) - (n-1)*(h+k choose k-1).
"As for A107027, it has a combinatorial interpretation: T(n,k) is the number of paths of length n*k such that each step has length 1 directed up or right and touches the line y = x/(n-1) only at lattice points.
"To see this, let us avoid notational confusion by replacing the "k" in A047089 by "j". Then the formula above becomes (h+j choose j) - (n-1)*(h+j choose j-1).
"If we sum over all the points at a distance n*k from (0,0) - i.e. if we sum from j=0 to j=k and let h = n*k-j - then we get (n*k choose k) - (n-2) * sum_{j=0}^{k-1} (n*k choose j) This is equivalent to the formula you report for A107027."

Examples

			Diagonals (beginning on row 0): {1}; {1,1}; {1,1,1}; {1,2,2,1};...

Crossrefs

See also the related array A107027.

Programs

  • PARI
    {T(n, k) = local(v); if( k<0 || k>n, 0, for(i=1, n+1, v=vector(i, j, if( j<2 || j>i-1, 1, v[j-1] + if( i%3 || i!=j+i\3, v[j])))); v[k+1])}; /* Michael Somos, Jan 28 2004 */
  • PARI
    {T(n, k) = if( k<0 || k>n, 0, if( n==0 && k==0, 1, T(n-1, k-1) + if( (n+1)%3 || n!=k+(n+1)\3, T(n-1, k))))}; /* Michael Somos, Jan 28 2004 */

Extensions

"Diagonals" in definition changed to "antidiagonals" by Michael Somos, Aug 19 2007

A107026 Row sums of inverse of Riordan array (1/(1+x),x/(1+x)^4).

Original entry on oeis.org

1, 2, 10, 62, 426, 3112, 23686, 185684, 1488554, 12144248, 100489320, 841268078, 7112138790, 60629940152, 520591221412, 4498091003272, 39079909924522, 341193986978008, 2991881019936760, 26338436818801496, 232688056611178216
Offset: 0

Views

Author

Paul Barry, May 09 2005

Keywords

Comments

The Riordan array (1/(1+x),x/(1+x)^4) has general term (-1)^(n-k)*binomial(n+3k,4k).

Crossrefs

Cf. A047098, A107027, A107030.

Programs

  • Maple
    A107026 := proc(n)
    3*binomial(4*n,n)-2*add(binomial(4*n,k),k=0..n) ;
end proc: # R. J. Mathar, Feb 20 2015

Formula

G.f.: A(x)=y satisfies (2y)^4*x-(y+1)^3*(y-1)=0.
a(n) = 3*binomial(4*n, n) - 2*Sum_{k=0..n} binomial(4*n, k).
Conjecture: +189*n*(3*n-1)*(3*n-2)*a(n) +72*(-1034*n^3+3098*n^2-3754*n+1655)*a(n
-1) +384*(2700*n^3-12828*n^2+20426*n-10785)*a(n-2) +4096*(-1066*n^3+6666*n^2-129
50*n+7365)*a(n-3) -65536*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Feb 20 2015
Conjecture: 3*n*(3*n-1)*(3*n-2)*(22*n^2-62*n+43)*a(n) +8*(-1892*n^5+8280*n^4-13330*n^3+9660*n^2-3048*n+315)*a(n-1) +128*(4*n-7)*(2*n-3)*(4*n-5)*(22*n^2-18*n+3)*a(n-2)=0. - R. J. Mathar, Feb 20 2015

A107030 Number triangle associated with the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 8, 6, 2, 1, 2, 16, 20, 8, 2, 1, 2, 32, 70, 38, 10, 2, 1, 2, 64, 252, 196, 62, 12, 2, 1, 2, 128, 924, 1062, 426, 92, 14, 2, 1, 2, 256, 3432, 5948, 3112, 792, 128, 16, 2, 1, 2, 512, 12870, 34120, 23686, 7302, 1326, 170, 18, 2, 1
Offset: 0

Views

Author

Paul Barry, May 09 2005

Keywords

Examples

			Triangle begins
  1;
  2,  1;
  2,  2,  1;
  2,  4,  2, 1;
  2,  8,  6, 2, 1;
  2, 16, 20, 8, 2, 1;

Crossrefs

Reversal of A107027.
Row sums are A107028.
Diagonal sums are A107031.
Columns include A040000, A000079, A000984, A047098.

Formula

Number triangle T(n, k)=(k-1)*C(k(n-k), n-k)-(k-2)*sum{j=0..n-k, C(k(n-k), j)}

A382100 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 35, 16, 1, 1, 1, 6, 31, 98, 126, 32, 1, 1, 1, 7, 46, 213, 531, 462, 64, 1, 1, 1, 8, 64, 396, 1556, 2974, 1716, 128, 1, 1, 1, 9, 85, 663, 3651, 11843, 17060, 6435, 256, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,     1,     1,     1, ...
  1,  1,   1,    1,     1,     1,     1, ...
  1,  2,   3,    4,     5,     6,     7, ...
  1,  4,  10,   19,    31,    46,    64, ...
  1,  8,  35,   98,   213,   396,   663, ...
  1, 16, 126,  531,  1556,  3651,  7391, ...
  1, 32, 462, 2974, 11843, 35232, 86488, ...

Crossrefs

Columns k=0..5 give A000012, A011782, A088218, A047099 (for n > 0), A107026(n)/2 (for n > 0), A304979(n)/2 (for n > 0).
Cf. A107027, A107030, A355262, A382101.

Programs

  • PARI
    a(n, k) = polcoef(1/(2-sum(j=0, n, binomial(k*j+1, j)/(k*j+1)*x^j+x*O(x^n))), n);
  • PARI
    a(n, k) = if(n==0, 1, (binomial(k*n, n)-(k-2)*sum(j=0, n-1, binomial(k*n, j)))/2);

Formula

A(n,k) = ( binomial(k*n,n) - (k-2) * Sum_{j=0..n-1} binomial(k*n,j) )/2 for n > 0.
G.f. of column k: 1/( 1 - Series_Reversion( x/(1+x)^k ) ).
Showing 1-6 of 6 results.

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