A091491 -id:A091491 - cp's OEIS frontend

A104402 Matrix inverse of triangle A091491, read by rows.

1, -1, 1, 1, -2, 1, 0, 2, -3, 1, 0, -1, 4, -4, 1, 0, 0, -3, 7, -5, 1, 0, 0, 1, -7, 11, -6, 1, 0, 0, 0, 4, -14, 16, -7, 1, 0, 0, 0, -1, 11, -25, 22, -8, 1, 0, 0, 0, 0, -5, 25, -41, 29, -9, 1, 0, 0, 0, 0, 1, -16, 50, -63, 37, -10, 1, 0, 0, 0, 0, 0, 6, -41, 91, -92, 46, -11, 1, 0, 0, 0, 0, 0, -1, 22, -91, 154, -129, 56, -12, 1

Offset: 0

Paul D. Hanna, Mar 05 2005

Row sums are all 0's for n>0. Absolute row sums form 2*A000045(n+1) for n>0, where A000045 = Fibonacci numbers. Sums of squared terms in row n = 2*A003440(n) for n>0, where A003440 = number of binary vectors with restricted repetitions.

Riordan array (1-x+x^2, x(1-x)). - Philippe Deléham, Nov 04 2009

			Triangle begins as:
   1;
  -1,  1;
   1, -2,  1;
   0,  2, -3,  1;
   0, -1,  4, -4,   1;
   0,  0, -3,  7,  -5,   1;
   0,  0,  1, -7,  11,  -6,  1;
   0,  0,  0,  4, -14,  16, -7,  1;
   0,  0,  0, -1,  11, -25, 22, -8, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000045, A003440, A091491, A199881.

Table[(-1)^(n-k)*(Binomial[k, n-k] + Binomial[k+1, n-k-1]), {n,0,12}, {k,0,n}] //Flatten (* G. C. Greubel, Apr 30 2021 *)

T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X^2)/(1-X*Y*(1-X)),n,x),k,y)

PARI
```
T(n,k)=if(n
				
```

T(n,k)=(-1)^(n-k)*(binomial(k,n-k)+binomial(k+1,n-k-1))

def A104402(n,k): return (-1)^(n+k)*(binomial(k,n-k) + binomial(k+1,n-k-1))
flatten([[A104402(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021

G.f.: (1-x+x^2)/(1-x*y*(1-x)).
T(n, k) = T(n-1, k-1) - T(n-2, k-1) for k>0 with T(0, 0)=1, T(1, 0)=-1, T(2, 0)=1, T(n, 0)=0 (n>2).
T(n, k) = (-1)^(n-k)*(C(k, n-k) + C(k+1, n-k-1)).
From Philippe Deléham, Nov 04 2009: (Start)
Sum_{k=0..n} T(n,k) = 0^n.
Sum_{k=0..n} abs(T(n, k)) = 2*Fibonacci(n+1) - [n=0].
Sum_{k=0..n} ( T(n,k) )^2 = 2*A003440(n) - [n=0]. (End)

A096465 Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 9, 1, 1, 5, 13, 22, 23, 1, 1, 6, 19, 41, 64, 65, 1, 1, 7, 26, 67, 131, 196, 197, 1, 1, 8, 34, 101, 232, 428, 625, 626, 1, 1, 9, 43, 144, 376, 804, 1429, 2055, 2056, 1, 1, 10, 53, 197, 573, 1377, 2806, 4861, 6917, 6918, 1, 1, 11, 64, 261, 834, 2211, 5017, 9878, 16795, 23713, 23714, 1

Offset: 0

Gerald McGarvey, Aug 12 2004

The third column is A034856 (binomial(n+1, 2) + n-1).
The row sums are A014137 (partial sums of Catalan numbers (A000108)).
The "1st subdiagonal" ((i+1,i) entries) are also A014137.
The "2nd subdiagonal" ((i+2,i) entries) is A014138 ( Partial sums of Catalan numbers (starting 1,2,5,...)).
The "3rd subdiagonal" ((i+3,i) entries) is A001453 (Catalan numbers - 1.)
This is the reverse of A091491 - see A091491 for more information. The sequence of antidiagonal sums gives A124642. - Gerald McGarvey, Dec 09 2006

			Triangle begins as:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  4,  1;
  1, 4,  8,  9,   1;
  1, 5, 13, 22,  23,   1;
  1, 6, 19, 41,  64,  65,   1;
  1, 7, 26, 67, 131, 196, 197, 1;

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A000108, A001453, A014137, A014138, A034856.

Cf. A006134, A024718, A030237, A078478, A091491, A100066, A105848, A124642.

a096465 n k = a096465_tabl !! n !! k
a096465_row n = a096465_tabl !! n
a096465_tabl = map reverse a091491_tabl
-- Reinhard Zumkeller, Jul 12 2012

A096465:= func< n,k | k eq n select 1 else (n-k)*(&+[Binomial(n+k-2*j, n-j)/(n+k-2*j): j in [0..k]]) >;
[A096465(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2021

A096465:= (n,k)-> `if`(k=n, 1, (n-k)*add(binomial(n+k-2*j, n-j)/(n+k-2*j), j=0..k));
seq(seq(A096465(n,k), k=0..n), n=0..12) # G. C. Greubel, Apr 30 2021

T[, 0]= 1; T[n, n_]= 1; T[n_, m_]:= T[n, m]= T[n-1, m] + T[n, m-1]; T[n_, m_] /; n < 0 || m > n = 0; Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2012 *)

def A096465(n,k): return 1 if (k==n) else (n-k)*sum( binomial(n+k-2*j, n-j)/(n+k-2*j) for j in (0..k))
flatten([[A096465(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021

From G. C. Greubel, Apr 30 2021: (Start)
T(n, k) = (n-k) * Sum_{j=0..k} binomial(n+k-2*j, n-j)/(n+k-2*j) with T(n,n) = 1.
T(n, k) = A091491(n, n-k).
Sum_{k=0..n} T(n,k) = Sum_{j=0..n} A000108(j) = A014137(n). (End)

Offset changed by Reinhard Zumkeller, Jul 12 2012

A099324 Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).

Original entry on oeis.org

1, 0, -1, 3, -8, 22, -64, 196, -625, 2055, -6917, 23713, -82499, 290511, -1033411, 3707851, -13402696, 48760366, -178405156, 656043856, -2423307046, 8987427466, -33453694486, 124936258126, -467995871776, 1757900019100, -6619846420552, 24987199492704, -94520750408708

Offset: 0

Paul Barry, Oct 12 2004

Binomial transform is A099323. Second binomial transform is A072100.

Hankel transform is A049347. - Paul Barry, Aug 10 2009

Robert Israel, Table of n, a(n) for n = 0..1668
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Cf. A014138.

f:= gfun:-rectoproc({(2+4*n)*a(n)+(4+5*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 0}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 27 2018

CoefficientList[Series[(1+Sqrt[1+4x])/(2(1+x)),{x,0,40}],x] (* Harvey P. Dale, Jan 30 2014 *)

a(n) = Sum_{k=0..2n} (2*0^(2n-k)-1)*C(k,floor(k/2)). - Paul Barry, Aug 10 2009
|a(n+2)| = A091491(n+2,2). - Philippe Deléham, Nov 25 2009
G.f.: T(0)/(2+2*x), where T(k) = k+2 - 2*x*(2*k+1) + 2*x*(k+2)*(2*k+3)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
D-finite with recurrence: (2+4*n)*a(n) + (4+5*n)*a(n+1) + (n+2)*a(n+2) = 0. - Robert Israel, Mar 27 2018

A185943 Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 16, 12, 5, 1, 6, 39, 34, 18, 6, 1, 7, 104, 98, 59, 25, 7, 1, 8, 301, 294, 190, 92, 33, 8, 1, 9, 927, 919, 618, 324, 134, 42, 9, 1, 10, 2983, 2974, 2047, 1128, 510, 186, 52, 10, 1, 11, 9901, 9891, 6908, 3934, 1887, 759, 249, 63, 11, 1

Offset: 0

Vladimir Kruchinin, Feb 07 2011

			Array begins
  1;
  2,   1;
  3,   3,   1;
  4,   7,   4,   1;
  5,  16,  12,   5,   1;
  6,  39,  34,  18,   6,   1;
  7, 104,  98,  59,  25,   7,   1;
  8, 301, 294, 190,  92,  33,   8,   1;
Production matrix begins:
   2, 1;
  -1, 1, 1;
   1, 1, 1, 1;
   0, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1;
   ... _Philippe Deléham_, Sep 20 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A091491 (m=1), A185944 (m=3), A185945 (m=4).

Row sums A014140. Cf. A000108, A014143.

r[n_, k_, m_] := k*Sum[ Binomial[i + m - 1, m - 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; r[n_, 0, 2] := n + 1; Table[r[n, k, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2012, from formula *)

@CachedFunction
def A(n, k):
    if n==k: return n+1
    return add(A(n-1, j) for j in (0..k))
A185943 = lambda n,k: A(n, n-k)
for n in (0..7) :
     print([A185943(n, k) for k in (0..n)])  # Peter Luschny, Nov 14 2012

R(n,k,m) = k*Sum_{i=0..n-k} binomial(i+m-1, m-1)*binomial(2*(n-i)-k-1, n-i-1)/(n-i), m = 2, k > 0.
R(n,0,2) = n + 1.
Conjecture: R(n,1,2) = A014140(n-1). R(n,2,2) = A014143(n-2). - R. J. Mathar, Feb 11 2011

A185944 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m = 3.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 10, 11, 5, 1, 15, 27, 17, 6, 1, 21, 66, 51, 24, 7, 1, 28, 170, 149, 83, 32, 8, 1, 36, 471, 443, 273, 124, 41, 9, 1, 45, 1398, 1362, 891, 448, 175, 51, 10, 1, 55, 4381, 4336, 2938, 1576, 685, 237, 62, 11, 1, 66, 14282, 14227, 9846, 5510, 2572, 996, 311, 74, 12, 1

Offset: 0

Vladimir Kruchinin, Feb 07 2011

			Array begins
   1;
   3,   1;
   6,   4,   1;
  10,  11,   5,   1;
  15,  27,  17,   6,   1;
  21,  66,  51,  24,   7,   1;
  28, 170, 149,  83,  32,   8,  1;
  36, 471, 443, 273, 124,  41,  9,   1;
Production matrix begins:
   3, 1;
  -3, 1, 1;
   4, 1, 1, 1;
  -2, 1, 1, 1, 1;
   1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1;
   ... _Philippe Deléham_, Sep 20 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A091491 (m=1), A185943 (m=2), A185945 (m=4), A014151 (column k=1).

Cf. A000108, A000217.

r[n_, k_, m_] := k*Sum[ Binomial[i + m - 1, m - 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; r[n_, 0, 3] = (n + 1)*(n + 2)/2; Table[ r[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

R(n,k,m) = k*Sum_(i=0..n-k,binomial(i+m-1,m-1)*binomial(2*(n-i)-k-1,n-i-1)/(n-i)), m=3, k>0.

R(n,0,3) = (n+1)*(n+2)/2 = A000217(n+1).

A185945 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m =4.

Original entry on oeis.org

1, 4, 1, 10, 5, 1, 20, 16, 6, 1, 35, 43, 23, 7, 1, 56, 109, 74, 31, 8, 1, 84, 279, 223, 114, 40, 9, 1, 120, 750, 666, 387, 164, 50, 10, 1, 165, 2148, 2028, 1278, 612, 225, 61, 11, 1, 220, 6529, 6364, 4216, 2188, 910, 298, 73, 12, 1, 286, 20811, 20591, 14062, 7698, 3482, 1294, 384, 86, 13, 1

Offset: 0

Vladimir Kruchinin, Feb 07 2011

			Array begins
    1;
    4,   1;
   10,   5,   1;
   20,  16,   6,   1;
   35,  43,  23,   7,   1;
   56, 109,  74,  31,   8,   1;
   84, 279, 223, 114,  40,   9,   1;
  120, 750, 666, 387, 164,  50,  10,   1;
Production matrix begins:
   4, 1;
  -6, 1, 1;
  10, 1, 1, 1;
  -9, 1, 1, 1, 1;
   7, 1, 1, 1, 1, 1;
  -3, 1, 1, 1, 1, 1, 1;
   1, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
   0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ... _Philippe Deléham_, Sep 20 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A091491 (m=1), A185943 (m=2), A185944 (m=3).

Cf. A000108, A000292.

r[n_, k_, m_] := k*Sum[ Binomial[i + m - 1, m - 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; r[n_, 0, 4] = Binomial[n + 3, 3]; Table[ r[n, k, 4], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

R(n,k,m) = k*Sum_{i=0..n-k} binomial(i+m-1, m-1)*binomial(2*(n-i)-k-1, n-i-1)/(n-i), m=4, k > 0.

R(n,0,4) = binomial(n+3,3) = A000292(n+1).

A155586 A modified Catalan sequence array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 2, 1, 1, 1, 14, 5, 2, 1, 1, 1, 42, 14, 5, 2, 1, 1, 1, 132, 42, 14, 5, 2, 1, 1, 1, 429, 132, 42, 14, 5, 2, 1, 1, 1, 1430, 429, 132, 42, 14, 5, 2, 1, 1, 1, 4862, 1430, 429, 132, 42, 14, 5, 2, 1, 1, 1, 16796, 4862, 1430, 429, 132, 42, 14, 5, 2, 1, 1

Offset: 0

Paul Barry, Jan 24 2009

Row sums are in A155587. Stieltjes associate array to A091491.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins
  1;
  1,    1;
  1,    1,   1;
  1,    2,   1,   1;
  1,    5,   2,   1,  1;
  1,   14,   5,   2,  1,  1;
  1,   42,  14,   5,  2,  1, 1;
  1,  132,  42,  14,  5,  2, 1, 1;
  1,  429, 132,  42, 14,  5, 2, 1, 1;
  1, 1430, 429, 132, 42, 14, 5, 2, 1, 1;
  ...

Cf. A000108, A091491, A155587.

T(n,k) = [k <= n] * if(k=0, 1, A000108(n-k)) for 0 <= k <= n, where [ ] is the Iverson bracket.

Bivariate o.g.f.: 1/(1 - x) + c(x) * x*y/(1 - x*y), where c(x) is the o.g.f. of A000108. - Petros Hadjicostas, Aug 03 2020

A168377 Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -1, 2, 1, 1, 1, 3, 4, 2, 1, -1, 11, 10, 7, 3, 1, 1, 31, 32, 21, 11, 4, 1, -1, 101, 100, 69, 37, 16, 5, 1, 1, 328, 329, 228, 128, 59, 22, 6, 1, -1, 1102, 1101, 773, 444, 216, 88, 29, 7, 1, 1, 3760, 3761, 2659, 1558, 785, 341, 125, 37, 8, 1

Offset: 0

Philippe Deléham, Nov 24 2009

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
   1;
  -1,   1;
   1,   0,   1;
  -1,   2,   1,  1;
   1,   3,   4,  2,  1;
  -1,  11,  10,  7,  3,  1;
   1,  31,  32, 21, 11,  4, 1;
  -1, 101, 100, 69, 37, 16, 5, 1;
  ...
From _Philippe Deléham_, Sep 14 2014: (Start)
Production matrix begins:
  -1, 1
   0, 1, 1
   0, 1, 1, 1
   0, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1, 1, 1
   0, 1, 1, 1, 1, 1, 1, 1, 1
   ... (End)

Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, Annals of Combinatorics, 13 (2009), 65-85.
L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, The Riordan group, Discrete Applied Mathematics, 34(1-3) (1991), 229-239.
Wikipedia, Riordan array.

Cf. A000012, A000108, A000124, A023443, A032357, A033297, A033999, A091491, A096470, A106566, A127540.

A000108(n) = binomial(2*n, n)/(n+1);
A032357(n) = sum(k=0, n, (-1)^(n-k)*A000108(k));
T(n, k) = if ((k==0), (-1)^n, if ((n<0) || (k<0), 0, if (k==1, A032357(n-1), if (n > k-1, T(n, k-1) - T(n-1, k-2), 0))));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020

T(n,0) = (-1)^n and T(n,n) = 1.
Sum_{0 <= k <= n} T(n,k) = A032357(n).
From Petros Hadjicostas, Aug 08 2020: (Start)
T(n,k) = T(n,k-1) - T(n-1,k-2) for 2 <= k <= n with initial conditions T(n,0) = (-1)^n (n >= 0) and T(n,1) = A032357(n-1) (n >= 1).
T(n,2) = A033297(n).
T(n,n-1) = n - 2 for n >= 1.
|T(n,k)| = |A096470(n,n-k)| for 0 <= k <= n.
Bivariate o.g.f.: 1/((1 + x)*(1 - x*y*c(x))), where c(x) is the o.g.f. of A000108.
Bivariate o.g.f.: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)).
Bivariate o.g.f. of |T(n,k)|: (o.g.f. of T(n,k)) + 2*x/(1 - x^2). (End)

cp's OEIS Frontend

A104402 Matrix inverse of triangle A091491, read by rows.

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A096465 Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).

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A099324 Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).

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A185943 Riordan array ((1/(1-x))^m, x*A000108(x)), m = 2.

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A185944 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m = 3.

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A185945 Riordan array ( (1/(1-x))^m , x*A000108(x) ), m =4.

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A155586 A modified Catalan sequence array.

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