A052509 -id:A052509 - cp's OEIS frontend

A131252 A052509 * A000012.

1, 2, 1, 4, 3, 1, 7, 6, 3, 1, 12, 11, 7, 3, 1, 20, 19, 14, 7, 3, 1, 33, 32, 26, 15, 7, 3, 1, 54, 53, 46, 30, 15, 7, 3, 1, 88, 87, 79, 57, 31, 15, 7, 3, 1, 143, 142, 133, 104, 62, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Left border = A000071, Fibonacci numbers - 1, starting with F(3): (1, 2, 4, 7, 12, ...). Row sums = A131253: (1, 3, 8, 17, 34, 64, 117, ...).

			First few rows of the triangle:
   1;
   2,  1;
   4,  3,  1;
   7,  6,  3,  1;
  12, 11,  7,  3,  1;
  20, 19, 14,  7,  3,  1;
  33, 32, 26, 15,  7,  3,  1;
  ...

Cf. A000071, A000012, A052509, A131253.

A052509 * A000012, where A000012 = (1; 1,1; 1,1,1; ...).

A131247 2*A052509 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 3, 1, 1, 7, 7, 3, 1, 1, 9, 13, 7, 3, 1, 1, 11, 21, 15, 7, 3, 1, 1, 13, 31, 29, 15, 7, 3, 1, 1, 15, 43, 51, 31, 15, 7, 3, 1, 1, 17, 57, 83, 61, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A104161 starting (1, 2, 5, 10, 19, 34, 59, ...). Reversal, A131248 is generated from 2*A004070 - A000012.

			First few rows of the triangle:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  3,  1;
  1,  7,  7,  3,  1;
  1,  9, 13,  7,  3,  1;
  1, 11, 21, 15,  7,  3,  1;
  ...

Cf. A052509, A104161, A004070, A131248.

2*A052509 - A000012, where A000012 = (1; 1,1; 1,1,1; ...).

A131249 A007318 * A052509.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 8, 12, 5, 1, 16, 32, 18, 6, 1, 32, 80, 57, 24, 7, 1, 64, 192, 168, 82, 31, 8, 1, 128, 448, 471, 260, 113, 39, 9, 1, 256, 1024, 1270, 790, 374, 152, 48, 10, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A061667: (1, 3, 9, 26, 73, 201, ...).

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,  1;
   8, 12,  5,  1;
  16, 32, 18,  6,  1;
  32, 80, 57, 24,  7,  1;
  ...

Cf. A061667, A007318, A052509.

Binomial transform of A052509.

A131251 A000012 * A052509.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 7, 3, 1, 6, 15, 14, 7, 3, 1, 7, 21, 25, 15, 7, 3, 1, 8, 28, 41, 30, 15, 7, 3, 1, 9, 36, 63, 56, 31, 15, 7, 3, 1, 10, 45, 92, 98, 62, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A001924: (1, 3, 7, 14, 26, 46, 79, ...). A131252 = A052509 * A000012.
From Clark Kimberling, Feb 07 2011: (Start)
When formatted as a rectangle R with northwest corner
  1, 2, 3,  4,  5,  6, ...
  1, 3, 6, 10, 15, 21, ...
  1, 3, 7, 14, 25, 41, ...
  1, 3, 7, 15, 30, 56, ...
  1, 3, 7, 15, 31, 62, ...
  ...
the following properties hold:
R is the accumulation array of the transpose of A052553 (a version of Pascal's triangle); see A144112 for the definition of accumulation array.
row 1: A000027
row 2: A000217
row 3: A004006
row 4: A055795
row 5: A057703
row 6: A115567
limiting row:  A000225
antidiagonal sums: A001924.
(End)

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  3,  1;
  5, 10,  7,  3,  1;
  6, 15, 14,  7,  3,  1;
  7, 21, 25, 15,  7,  3,  1;
  ...

Cf. A052509, A000012, A001924, A131252.

A000012 * A052509 as infinite lower triangular matrices.

A131256 A000012(signed) * A052509.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 3, 1, 1, 0, 3, 4, 3, 1, 1, 1, 3, 7, 5, 3, 1, 1, 0, 4, 9, 10, 5, 3, 1, 11, 4, 13, 16, 11, 5, 3, 1, 1, 0, 5, 16, 26, 20, 11, 5, 3, 1, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A052952: (1, 1, 3, 4, 8, 12, 21, 33, ...). A131257 = A052509 * A000012(signed).

			First few rows of the triangle:
  1;
  0, 1;
  1, 1, 1;
  0, 2, 1, 1;
  1, 2, 3, 1, 1;
  0, 3, 4, 3, 1, 1;
  1, 3, 7, 5, 3, 1, 1;
  ...

Cf. A052509, A000012, A131257.

A000012(signed) * A052509, where the signed version of A000012 = (1; -1,1; 1,-1,1; ...).

A131257 Triangle read by rows: A052509 * A097807 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, -1, 2, 1, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 1, 0, 6, 5, 3, 1, 1, 0, 1, 6, 10, 5, 3, 1, 1, 0, 1, 7, 15, 11, 5, 3, 1, 1, -1, 2, 7, 22, 20, 11, 5, 3, 1, 1, 0, 1, 9, 28, 36, 21, 11, 5, 3, 1, 1, 0, 1, 10, 36, 57, 42, 21, 11, 5, 3, 1, 1, 1, 0, 12, 44, 86, 77, 43, 21, 11, 5, 3, 1, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums give A099517.

			First few rows of the triangle:
   1;
   0, 1;
   0, 1, 1;
  -1, 2, 1, 1;
   0, 1, 3, 1, 1;
   0, 1, 4, 3, 1, 1;
   1, 0, 6, 5, 3, 1, 1;
  ...

Cf. A099517, A052509, A097807, A131256.

A052509 * A000012 (signed, + - + - by columns).

a(48) = 22 corrected and more terms from Georg Fischer, Jun 05 2023

A141905 Triangle read by rows, T(n, k) = binomial(n, k)*A052509(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 24, 8, 1, 1, 25, 70, 40, 10, 1, 1, 36, 165, 160, 60, 12, 1, 1, 49, 336, 525, 280, 84, 14, 1, 1, 64, 616, 1456, 1120, 448, 112, 16, 1, 1, 81, 1044, 3528, 3906, 2016, 672, 144, 18, 1, 1, 100, 1665, 7680, 11970, 8064, 3360, 960, 180, 20, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

Original definition: A skew trinomial summed triangular sequence of coefficients: T(n, k) = Sum_{j=0..k} n!/((n - k - j)!*j!*k!).
It is obscure how the above formula is used for the region where the sum reaches k > n-m, which needs a definition of the factorials at negative integer argument. If we trust the author's Mma implementation, Mma throws in some magic renormalization to cover these arguments. If we define, properly, t(n, k) = Sum_{j=0..n-k} n!/((n-k-j)!*j!*k!), then we recover just A038207. - R. J. Mathar, Feb 07 2014
Let p(n, k, j) = n!/((n-k-j)!*j!*k!), for j<=n-k and 0<= k <=n and p(n, k, j) = 0, for j > n-k and 0<= k <=n. It seems that T(n, k) coincides with Sum_{j=0..k} p(n, k, j). - Luis Manuel Rivera Martínez, Mar 04 2014

			Triangle begins as:
[0]  1;
[1]  1,   1;
[2]  1,   4,    1;
[3]  1,   9,    6,    1;
[4]  1,  16,   24,    8,     1;
[5]  1,  25,   70,   40,    10,    1;
[6]  1,  36,  165,  160,    60,   12,    1;
[7]  1,  49,  336,  525,   280,   84,   14,   1;
[8]  1,  64,  616, 1456,  1120,  448,  112,  16,   1;
[9]  1,  81, 1044, 3528,  3906, 2016,  672, 144,  18,  1;

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

Row sums are A027914.

Cf. A038207, A052509.

[Binomial(n,k)*(&+[Binomial(n-k,j): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 29 2021

A052509 := proc(n, k) option remember: if k = 0 or k = n then 1 else A052509(n-1, k) + A052509(n-2, k-1) fi end: T := (n, k) -> binomial(n, k)*A052509(n, k): seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Nov 26 2021

T[n_, k_]:= Sum[n!/((n-k-j)!*j!*k!), {j,0,k}];
Table[T[n, k], {n, 0, 10}, {k,0,n}] // Flatten

flatten([[binomial(n,k)*sum(binomial(n-k,j) for j in (0..k)) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 29 2021

T(n, k) = Sum_{j=0..k} n!/((n - k - j)!*j!*k!).
G.f.: (2*x)/((3*x - 1)*sqrt(-4*x^2*y + x^2 - 2*x + 1) - 4*x^2*y + x^2 - 2*x +1). - Vladimir Kruchinin, Oct 05 2020
T(n, k) = binomial(n, k)*hypergeom([-k, -n + k], [-k], -1). - Peter Luschny, Nov 28 2021

Edited by G. C. Greubel, Mar 29 2021

New name by Peter Luschny, Nov 26 2021

A131258 A129686^(-1) * A052509.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 3, 2, 1, 0, 4, 8, 6, 3, 2, 1, 0, 4, 11, 12, 6, 3, 2, 1, 1, 4, 14, 20, 13, 6, 3, 2, 1, 1, 5, 18, 30, 25, 13, 6, 3, 2, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A097083: (1, 2, 3, 5, 9, 15, ...).

			First few rows of the triangle:
  1;
  1, 1;
  0, 2, 1;
  0, 2, 2, 1;
  1, 2, 3, 2, 1;
  1, 3, 5, 3, 2, 1;
  0, 4, 8, 6, 3, 2, 1;
  ...

Cf. A129686, A052509, A097083.

A129686^(-1) * A052509 as infinite lower triangular matrices, where A129686 = the alternate term operator.

A004070 Table of Whitney numbers W(n,k) read by antidiagonals, where W(n,k) is maximal number of pieces into which n-space is sliced by k hyperplanes, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 2, 4, 7, 5, 1, 1, 2, 4, 8, 11, 6, 1, 1, 2, 4, 8, 15, 16, 7, 1, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 1, 2, 4, 8, 16, 32, 64, 120, 163

Offset: 0

N. J. A. Sloane

As a number triangle, this is given by T(n,k)=sum{j=0..n, C(n,j)(-1)^(n-j)sum{i=0..j, C(j+k,i-k)}}. - Paul Barry, Aug 23 2004
As a number triangle, this is the Riordan array (1/(1-x), x(1+x)) with T(n,k)=sum{i=0..n, binomial(k,i-k)}. Diagonal sums are then A023434(n+1). - Paul Barry, Feb 16 2005
Form partial sums across rows of square array of binomial coefficients A026729; see also A008949. - Philippe Deléham, Aug 28 2005
Square array A026729 -> Partial sums across rows
1 0 0 0 0 0 0 . . . . 1 1 1 1 1 1 1 . . . . . .
1 1 0 0 0 0 0 . . . . 1 2 2 2 2 2 2 . . . . . .
1 2 1 0 0 0 0 . . . . 1 3 4 4 4 4 4 . . . . . .
1 3 3 1 0 0 0 . . . . 1 4 7 8 8 8 8 . . . . . .
For other Whitney numbers see A007799.
W(n,k) is the number of length k binary sequences containing no more than n 1's. - Geoffrey Critzer, Mar 15 2010
From Emeric Deutsch, Jun 15 2010: (Start)
Viewed as a number triangle, T(n,k) is the number of internal nodes of the Fibonacci tree of order n+2 at level k. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
(End)
Named after the American mathematician Hassler Whitney (1907-1989). - Amiram Eldar, Jun 13 2021

			Table W(n,k) begins:
  1 1 1 1  1  1  1 ...
  1 2 3 4  5  6  7 ...
  1 2 4 7 11 16 22 ...
  1 2 4 8 15 26 42 ...
W(2,4) = 11 because there are 11 length 4 binary sequences containing no more than 2 1's: {0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}, {0, 0, 1, 1}, {0, 1, 0, 0}, {0, 1, 0, 1}, {0, 1, 1, 0}, {1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 1, 0, 0}. - _Geoffrey Critzer_, Mar 15 2010
Table T(n, k) begins:
  1
  1  1
  1  2  1
  1  2  3  1
  1  2  4  4  1
  1  2  4  7  5  1
  1  2  4  8 11  6  1
...

Donald E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Gustav Burosch, Hans-Dietrich O.F. Gronau, Jean-Marie Laborde and Ingo Warnke, On posets of m-ary words, Discrete Math., Vol. 152, No. 1-3 (1996), pp. 69-91. MR1388633 (97e:06002)
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Matteo Cervetti and Luca Ferrari, Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset, Annals of Combinatorics (2022).
Richard K. Guy, Letter to N. J. A. Sloane, Apr 1975.
Yasuichi Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, Vol. 20, No. 2 (1982), pp. 168-178.
Robin Pemantle and Mark C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., Vol. 50, No. 2 (2008), pp. 199-272. See p. 233.

Cf. A007799. As a triangle, mirror A052509.
Rows converge to powers of two (A000079). Subdiagonals include A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, A035042. Antidiagonal sums are A000071.
Rows are: A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863.
Cf. A178522, A178524.

Transpose[ Table[Table[Sum[Binomial[n, k], {k, 0, m}], {m, 0, 15}], {n, 0, 15}]] // Grid (* Geoffrey Critzer, Mar 15 2010 *)
T[ n_, k_] := Sum[ Binomial[n, j] (-1)^(n - j) Sum[ Binomial[j + k, i - k], {i, 0, j}], {j, 0, n}]; (* Michael Somos, May 31 2016 *)

/* array read by antidiagonals up coordinate index functions */
t1(n) = binomial(floor(3/2 + sqrt(2+2*n)), 2) - (n+1); /* A025581 */
t2(n) = n - binomial(floor(1/2 + sqrt(2+2*n)), 2); /* A002262 */
/* define the sequence array function for A004070 */
W(n, k) = sum(i=0, n, binomial(k, i));
/* visual check ( origin 0,0 ) */
printp(matrix(7, 7, n, k, W(n-1, k-1)));
/* print the sequence entries by antidiagonals going up ( origin 0,0 ) */
print1("S A004070 "); for(n=0, 32, print1(W(t1(n), t2(n))","));
print1("T A004070 "); for(n=33, 61, print1(W(t1(n), t2(n))","));
print1("U A004070 "); for(n=62, 86, print1(W(t1(n), t2(n))",")); /* Michael Somos, Apr 28 2000 */

T(n, k)=sum(m=0, n-k, binomial(k, m)) \\ Jianing Song, May 30 2022

W(n, k) = Sum_{i=0..n} binomial(k, i). - Bill Gosper
W(n, k) = if k=0 or n=0 then 1 else W(n, k-1)+W(n-1, k-1). - David Broadhurst, Jan 05 2000
The table W(n,k) = A000012 * A007318(transform), where A000012 = (1; 1,1; 1,1,1; ...). - Gary W. Adamson, Nov 15 2007
E.g.f. for row n: (1 + x + x^2/2! + ... + x^n/n!)* exp(x). - Geoffrey Critzer, Mar 15 2010
G.f.: 1 / (1 - x - x*y*(1 - x^2)) = Sum_{0 <= k <= n} x^n * y^k * T(n, k). - Michael Somos, May 31 2016
W(n, n) = 2^n. - Michael Somos, May 31 2016
From Jianing Song, May 30 2022: (Start)
T(n, 0) = T(n, n) = 1 for n >= 0; T(n, k) = T(n-1, k-1) + T(n-2, k-1) for k=1, 2, ..., n-1, n >= 2.
T(n, k) = Sum_{m=0..n-k} binomial(k, m).
T(n,k) = 2^k for 0 <= k <= floor(n/2). (End)

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000

A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 0, 1, 12, 55

Offset: 0

N. J. A. Sloane, Mar 17 2000

Another version of Pascal's triangle A007318.
As a triangle read by rows, it is (1,0,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938 and it is the Riordan array (1/(1-x), x^2/(1-x)). The row sums of this triangle are F(n+1) = A000045(n+1). - Philippe Deléham, Dec 11 2011
As a triangle, binomial(n-k, k) is also the number of ways to add k pierced circles to a path graph P_n so that no two circles share a vertex (see Lemma 3.1 at page 5 in Owad and Tsvietkova). - Stefano Spezia, May 18 2022
For all n >= 0, k >= 0, the k-th homology group of the n-torus H_k(T^n) is the free abelian group of rank T(n,k) = binomial(n,k). See the Math Stack Exchange link below. - Jianing Song, Mar 13 2023

			Array begins:
  1, 0,  0,  0, 0, 0, ...
  1, 1,  0,  0, 0, 0, ...
  1, 2,  1,  0, 0, 0, ...
  1, 3,  3,  1, 0, 0, ...
  1, 4,  6,  4, 1, 0, ...
  1, 5, 10, 10, 5, 1, ...
As a triangle, this begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  0, 0;
  1, 3,  1, 0, 0;
  1, 4,  3, 0, 0, 0;
  1, 5,  6, 1, 0, 0, 0;
  1, 6, 10, 4, 0, 0, 0, 0;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5459
Math Stack Exchange, Homology of the n-torus using the Künneth Formula
Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.
Index entries for triangles and arrays related to Pascal's triangle

The official entry for Pascal's triangle is A007318. See also A026729 (the same array read by downward antidiagonals).
As a triangle without zeros: A011973.
Cf. A052509, A054123, A054124, A008949.

/* As triangle */ [[Binomial(n-k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 08 2017

with(combinat): for s from 0 to 20 do for n from s to 0 by -1 do printf(`%d,`, binomial(n, s-n)) od:od: # James Sellers, Mar 17 2000

Flatten[ Table[ Binomial[n-k , k], {n, 0, 13}, {k, 0, n}]]  (* Jean-François Alcover, Dec 05 2012 *)

T(n,k) = binomial(n,k) \\ Charles R Greathouse IV, Feb 07 2017

As a triangle: T(n,k) = A026729(n,n-k).

G.f. of the triangular version: 1/(1-x-x^2*y). - R. J. Mathar, Aug 11 2015

cp's OEIS Frontend

A131252 A052509 * A000012.

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A131247 2*A052509 - A000012.

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A131249 A007318 * A052509.

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A131251 A000012 * A052509.

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A131256 A000012(signed) * A052509.

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A131257 Triangle read by rows: A052509 * A097807 as infinite lower triangular matrices.

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A141905 Triangle read by rows, T(n, k) = binomial(n, k)*A052509(n, k) for 0 <= k <= n.

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Magma

Maple

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A131258 A129686^(-1) * A052509.

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A004070 Table of Whitney numbers W(n,k) read by antidiagonals, where W(n,k) is maximal number of pieces into which n-space is sliced by k hyperplanes, n >= 0, k >= 0.

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