A107030 -id:A107030 - cp's OEIS frontend

A107031 Diagonal sums of number triangle A107030.

1, 2, 3, 4, 7, 12, 25, 56, 145, 422, 1389, 5072, 20283, 87996, 412663, 2087868, 11361831, 66155642, 409567833, 2680486870, 18465061963, 133504614880, 1011121118741, 8008145020804, 66201411633209, 570006877641976

Offset: 0

Paul Barry, May 09 2005

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

A107027 Number triangle associated to the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

Original entry on oeis.org

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

Offset: 0

Paul Barry, May 09 2005

As a number square read by antidiagonals, the rows represent the row sums of the inverses of the Riordan arrays (1/(1+x),x/(1+x)^k), k>=0. The rows are then given by T(n,k)=(n-1)C(n*k,k)-(n-2)*sum{j=0..k, C(n*k,j)}.

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

T(n,n) is A040000, T(n+1,n) is A000079, T(n+2,n) is A000984, T(n+3,n) is A047098.
The reverse of this triangle is A107030.
Row sums are A107028.
Diagonal sums are A107029.

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

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

Paul Barry, May 09 2005

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

Cf. A047098, A107027, A107030.

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

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

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

Seiichi Manyama, Mar 15 2025

			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, ...

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.

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);

a(n, k) = if(n==0, 1, (binomial(k*n, n)-(k-2)*sum(j=0, n-1, binomial(k*n, j)))/2);

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 ) ).

cp's OEIS Frontend

A107031 Diagonal sums of number triangle A107030.

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A107027 Number triangle associated to the Riordan arrays (1/(1+x),x/(1+x)^k),k>=0.

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

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Maple

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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.

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PARI

PARI

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