A007318 -id:A007318 - cp's OEIS frontend

A131056 A007318 * A131055.

1, 3, 7, 17, 41, 97, 225, 513, 1153, 2561, 5633, 12289, 26625, 57345, 122881, 262145, 557057, 1179649, 2490369, 5242881, 11010049, 23068673, 48234497, 100663297, 209715201, 436207617, 905969665

Offset: 1

Gary W. Adamson, Jun 12 2007

			a(4) = 17 = (1, 3, 3, 1) dot (1, 2, 2, 4) = (1 + 6 + 6 + 4).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4)

Cf. A131055, A131054, A141222.

LinearRecurrence[{5,-8,4},{1,3,7,17},40] (* Harvey P. Dale, Apr 30 2022 *)

Vec(x*(1-2*x+2*x^3)/((1-x)*(1-2*x)^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

Binomial transform of A131055: (1, 2, 2, 4, 4, 6, 6, ...). A131056 = A131054 as an infinite lower triangular matrix * [1,2,3,...] as a vector.
G.f.: x*(1-2*x+2*x^3)/((1-x)*(1-2*x)^2); a(n)=-0^n/2+2^(n-1)*(n+1)+1. - Paul Barry, Jun 14 2008
a(n) = 2+A099035(n-1), n>1. - Juri-Stepan Gerasimov, Oct 02 2011

More terms from Paul Barry, Jun 14 2008

A132047 3A007318 - 2A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 9, 9, 1, 1, 12, 18, 12, 1, 1, 15, 30, 30, 15, 1, 1, 18, 45, 60, 45, 18, 1, 1, 21, 63, 105, 105, 63, 21, 1, 1, 24, 84, 168, 210, 168, 84, 24, 1, 1, 27, 108, 252, 378, 378, 252, 108, 27, 1, 1, 30, 135, 360, 630, 756, 630, 360, 135, 30, 1

Offset: 0

Gary W. Adamson, Aug 08 2007

			First few rows of the triangle are:
  1;
  1, 1;
  1, 6, 1;
  1, 9, 9, 1;
  1, 12, 18, 12, 1;
  1, 15, 30, 30, 15, 1;
  1, 18, 45, 60, 45, 18, 1;
  ...

Cf. A007318, A103451, A131128 (row sums).

T(n, k) = my(bnk = binomial(n, k)); 3*bnk - 2*(bnk==1); \\ Michel Marcus, Jun 16 2022

a(n) = 3*A007318(n) - 2*A103451(n).

T(n,k) = 3*C(n,k)-2*(C(n,k-n)+C(n,-k)-C(0,n+k)), 0<=k<=n. [Eric Werley, Jul 01 2011]

Corrected and extended by Roger L. Bagula, Nov 02 2008

A152845 Triangle read by rows, A007318 rows repeated seven times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1

Offset: 0

Philippe Deléham, Dec 14 2008

Diagonal sums : A103375 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

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

Cf. A007318, A152198, A152815, A152828, A152830, A152831, A152844

Flatten[Table[#, {7}] & /@ Table[Binomial[n, k], {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

A175807 A007318-perfect numbers.

Original entry on oeis.org

2, 3, 4, 5, 12, 22, 26, 154

Offset: 1

Vladimir Shevelev, Dec 05 2010

See definition in comment to A175522. The definition is applied to the flattened view of the binomial coefficients with a single index, without regard to fact that A007318 is a triangle.

No more terms up to 10^6. - Michel Marcus, Feb 07 2016

			Since A007318(1)+ A007318(2)+ A007318(3)+ A007318(4)+ A007318(6)=6= A007318(12), then 12 is in the sequence.

Cf. A175811, A175522, A000396.

A007318 := proc(n) option remember; local t,r; t := 0 ; for r from 0 do if t+r+1 > n then return binomial(r,n-t) ; end if; t := t+r+1 ; end do: end proc:
isA175807 := proc(n) m := 0 ; for d in numtheory[divisors](n) minus {n} do m := m+A007318(d) ; end do; m = A007318(n) ; end proc:
for n from 1 do if isA175807(n) then printf("%d,\n",n); end if; end do: # R. J. Mathar, Dec 05 2010

b(n) = {my(m = 1); while (m*(m+1)/2 < n, m++); if (! ispolygonal(n, 3), m--); binomial(m, n - m*(m+1)/2);}
isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 07 2016

A175811 A007318-deficient numbers.

Original entry on oeis.org

1, 7, 11, 13, 17, 18, 19, 23, 24, 25, 29, 30, 31, 32, 33, 37, 38, 39, 40, 41, 42, 43, 47, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 93, 94, 95, 96, 97, 98, 99, 100, 101, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117

Offset: 1

Vladimir Shevelev, Dec 05 2010

Definition see in comment to A175522. The same criticism on index-selection as in A175807 applies. All primes greater than 5 are in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007318, A175522, A175807 (perfect version), A005100, A005101.

A007318 := proc(n) option remember; local t, r; t := 0 ; for r from 0 do if t+r+1 > n then return binomial(r, n-t) ; end if; t := t+r+1 ; end do: end proc:
isA175811 := proc(n) m := 0 ; for d in numtheory[divisors](n) minus {n} do m := m+A007318(d) ; end do; m < A007318(n) ; end proc:
for n from 1 to 120 do if isA175811(n) then printf("%d,", n); end if; end do: # R. J. Mathar, Dec 06 2010

b(n) = {my(m = 1); while (m*(m+1)/2 < n, m++); if (! ispolygonal(n, 3), m--); binomial(m, n - m*(m+1)/2);}
isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 07 2016

{n: sum_{d|n, dA007318(d) < A007318(n)}.

Terms >25 from R. J. Mathar, Dec 06 2010

A055375 Euler transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 14, 21, 14, 5, 7, 26, 48, 48, 26, 7, 11, 45, 103, 131, 103, 45, 11, 15, 75, 198, 312, 312, 198, 75, 15, 22, 120, 366, 674, 830, 674, 366, 120, 22, 30, 187, 637, 1359, 1961, 1961, 1359, 637, 187, 30, 42, 284, 1078, 2584, 4302, 5066, 4302, 2584, 1078, 284, 42

Offset: 0

Christian G. Bower, May 16 2000

Number of partitions of n objects, k of which are black, into parts each of which is a sequence of objects. E.g. T(3,1) = 7; the partitions are [BWW], [WBW], [WWB], [BW,W], [WB,W], [WW,B] and [B,W,W]. - Franklin T. Adams-Watters, Jan 10 2007

			Triangle begins
   1;
   1,  1;
   2,  3,   2;
   3,  7,   7,   3;
   5, 14,  21,  14,   5;
   7, 26,  48,  48,  26,   7;
  11, 45, 103, 131, 103,  45, 11;
  15, 75, 198, 312, 312, 198, 75, 15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A034899.
Columns k=0-1 give: A000041, A014153(n-1) for n>=1.
T(2n,n) gives A360626.
Cf. A007318, A034691, A209406, A360634.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 14 2023

nmax = 10; pp = Product[Product[1/(1 - x^i*y^j)^Binomial[i, j], {j, 0, i}], {i, 1, nmax}]; t[n_, k_] := SeriesCoefficient[pp, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2017 *)

G.f.: Product_{i>=1} Product_{j=0..i} 1/(1 - x^i y^j)^C(i,j). - Franklin T. Adams-Watters, Jan 10 2007

Sum_{k=0..2n} (-1)^k * T(2n,k) = A034691(n). - Alois P. Heinz, Dec 05 2023

A073617 Consider Pascal's triangle A007318; a(n) = product of terms at +45 degrees slope with the horizontal.

Original entry on oeis.org

1, 1, 1, 2, 3, 12, 30, 240, 1050, 16800, 132300, 4233600, 61122600, 3911846400, 104886381600, 13425456844800, 674943865596000, 172785629592576000, 16407885372638760000, 8400837310791045120000, 1515727634953623371280000, 1552105098192510332190720000

Offset: 0

Amarnath Murthy, Aug 07 2002

nonn

The sum of the terms pertaining to the above product is the (n+1)-th Fibonacci number: 1 + 5 + 6 + 1 = 13.

n divides A073617(n+1) for n>=1; see the Mathematica section. [Clark Kimberling, Feb 29 2012]

			For n=6 the diagonal is 1,5,6,1 and product of the terms = 30 hence a(6) = 30.

Vaclav Kotesovec, Table of n, a(n) for n = 0..118

Cf. A000045, A073618, A007685, A208649, A000984.

a:= n-> mul(binomial(n-i, i), i=0..floor(n/2)):
seq(a(n), n=0..21);  # Alois P. Heinz, Nov 27 2023

p[n_] := Product[Binomial[n + 1 - k, k], {k, 1, Floor[(n + 1)/2]}]
Table[p[n], {n, 1, 20}]   (* A073617(n+1) *)
Table[p[n]/n, {n, 1, 20}] (* A208649 *)
(* Clark Kimberling, Feb 29 2012 *)
(* Second program *)
Join[{1}, Table[If[EvenQ[n], 2^(3/2 - n/4) * Sqrt[BarnesG[n]] * Gamma[n] / (n*BarnesG[n/2]^2 * Gamma[n/2]^(7/2)), Glaisher^3 * 2^((-10 + 3*n + 6*n^2)/12) * BarnesG[n/2]^2 * Gamma[n/2]^(5/2) / (E^(1/4) * Pi^(1/4 + n/2) * Sqrt[BarnesG[n]] * Gamma[n])], {n, 1, 25}]] (* Vaclav Kotesovec, Jun 10 2025 *)

a(n) = Product_{k=0..floor(n/2)} binomial(n-k,k).
a(2n+1)/a(2n-1) = binomial(2n,n); a(2n)/a(2n-2) = (1/2)*binomial(2n,n); (a(2n+1)*a(2n-2))/(a(2n)*a(2n-1)) = 2. - John Molokach, Sep 09 2013
a(n) ~ A^(3/2) * 2^(n*(n+1)/4 - 1/6 + (-1)^n/4) * exp(n/4 - 1/8) / (n^((n+1)/4 + (-1)^n/8) * Pi^(n/4 + 3/8 + (-1)^n/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jun 10 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 22 2003

A080233 Triangle T(n,k) obtained by taking differences of consecutive pairs of row elements of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 0, 1, 1, -1, 1, 2, 0, -2, 1, 3, 2, -2, -3, 1, 4, 5, 0, -5, -4, 1, 5, 9, 5, -5, -9, -5, 1, 6, 14, 14, 0, -14, -14, -6, 1, 7, 20, 28, 14, -14, -28, -20, -7, 1, 8, 27, 48, 42, 0, -42, -48, -27, -8, 1, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9

Offset: 0

Paul Barry, Feb 10 2003

Row sums are 1,1,1,1,1,1 with g.f. 1/(1-x). Can also be obtained from triangle A080232 by taking sums of pairs of consecutive row elements.

Mirror image of triangle in A156644. - Philippe Deléham, Feb 14 2009

			Triangle begins as:
  1;
  1, 0;
  1, 1, -1;
  1, 2,  0, -2;
  1, 3,  2, -2, -3;
  1, 4,  5,  0, -5,  -4;
  1, 5,  9,  5, -5,  -9,  -5;
  1, 6, 14, 14,  0, -14, -14,  -6;
  1, 7, 20, 28, 14, -14, -28, -20,  -7;
  1, 8, 27, 48, 42,   0, -42, -48, -27,  -8;
  1, 9, 35, 75, 90,  42, -42, -90, -75, -35, -9;
  ...

Row sums give A000012.

Cf. A007318, A080232.

Table[Binomial[n, k] - Binomial[n, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

{T(n, k) = if( n<0 || k>n, 0, binomial(n, k) - binomial(n, k-1))}; /* Michael Somos, Nov 25 2016 */

T(n, k) = if(k>n, 0, binomial(n, k)-binomial(n, k-1)).

A091044 One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.

Original entry on oeis.org

1, 2, 2, 3, 10, 3, 4, 28, 28, 4, 5, 60, 126, 60, 5, 6, 110, 396, 396, 110, 6, 7, 182, 1001, 1716, 1001, 182, 7, 8, 280, 2184, 5720, 5720, 2184, 280, 8, 9, 408, 4284, 15912, 24310, 15912, 4284, 408, 9, 10, 570, 7752, 38760, 83980, 83980, 38760, 7752, 570, 10, 11

Offset: 1

Wolfdieter Lang, Jan 23 2004

The odd-numbered columns of this triangle can be reduced: see triangle A091043.
The odd-numbered rows coincide with the ones of the reduced triangle A091043.
binomial(2*n,2*m+1) is even for n >= m + 1 >= 1, hence every T(n,m) is a positive integer.
The GCD (greatest common divisor) of the entries of each odd-numbered row n=2*k+1, k>=0, is 1.
The GCD of the entries of the even-numbered row n=2*k, k>=1, is A006519(n) (highest power of 2 in n=2*k).

			Triangle begins:
  [1];
  [2,2];
  [3,10,3];
  [4,28,28,4];
  [5,60,126,60,5];
  [6,110,396,396,110,6];
  ...
n = 6 = 2*3: gcd(6,110,396) = 2 = A006519(6);
n = 5: gcd(5,60,126) = 1 = A006519(5).

Indranil Ghosh, Rows 1..125, flattened
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 4.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.
Hernan de Alba, W. Carballosa, J. Leaños, and L. M. Rivera, Independence and matching numbers of some token graphs, arXiv preprint arXiv:1606.06370 [math.CO], 2016.
Wolfdieter Lang, First 9 rows.

Cf. A000302 (row sums), A001109, A006519, A007318, A053124, A091042, A091043, A111910.

Flatten[Table[Binomial[2n,2m+1]/2,{n,1,11},{m,0,n-1}]] (* Indranil Ghosh, Feb 22 2017 *)

{A(i, j) = binomial(2*i + 2*j - 2, 2*i - 1) / 2}; /* Michael Somos, Oct 15 2017 */

T(n, m)= binomial(2*n, 2*m+1)/2, n >= m + 1 >= 1, else 0.
  Put a(n) = n!*(n+1/2)!/(1/2)!. T(n+1,k) = (n+1)*a(n)/(a(k)*a(n-k)).
  T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1). Cf. A111910. - Peter Bala, Oct 13 2011
From Peter Bala, Jul 29 2013: (Start)
O.g.f.: 1/(1 - 2*t*(x + 1) + t^2*(x - 1)^2)= 1 + (2 + 2*x)*t + (3 + 10*x + 3*x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(4*sqrt(x))*( (1 + sqrt(x))^(2*n) - (sqrt(x) - 1)^(2*n) ) and has n-1 real zeros given by the formula -cot^2(k*Pi/(2*n)) for k = 1,2,...,n-1. Cf A091042.
The row polynomial R(n,x) satisfies (x - 1)^n*R(n,x/(x - 1)) = U(n,2*x - 1), the n-th row polynomial of A053124.
Row sums A000302. Sum {k = 0..n-1} 2^k*T(n,k) = A001109(n). (End)
From Werner Schulte, Jan 13 2017: (Start)
(1) T(n,m) = T(n-1,m) + T(n-1,m-1)*(2*n-1-m)/m for 0 < m < n-1 with T(n,0) = n and T(n,n) = 0;
(2) T(n,m) = 2*T(n-1,m) + 2*T(n-1,m-1) - T(n-2,m) + 2*T(n-2,m-1) - T(n-2,m-2) for 0 < m < n-1 with T(n,0) = T(n,n-1) = n and T(n,m) = 0 if m < 0 or m >= n;
(3) The row polynomials p(n,x) = Sum_{m=0..n-1} T(n,m)*x^m satisfy the recurrence equation p(n+2,x) = (2+2*x)*p(n+1,x) - (x-1)^2*p(n,x) for n >= 1 with initial values p(1,x) = 1 and p(2,x) = 2+2*x.
(End)
G.f.: x*y /(1 - 2*(x+y) + (x-y)^2) with the entries regarded as an infinite square array A(i, j) read by antidiagonals. - Michael Somos, Oct 15 2017

A116672 Triangle read by rows in which the binomial transform of the n-th row gives the Euler transform of the n-th diagonal of Pascal's triangle (A007318).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 29, 12, 1, 1, 14, 57, 96, 72, 21, 1, 1, 21, 117, 277, 319, 176, 38, 1

Offset: 1

Alford Arnold, Feb 22 2006

For example, the Euler transform of 1,3,6,... is 1,1,4,10,26,59,141,... (A000294) differing slightly from A000293 which counts the solid partitions.
The NAME does not reproduce the DATA, COMMENTS, or EXAMPLES.  - R. J. Mathar, Jul 19 2017
The binomial transforms of the rows form the rows of A289656. - N. J. A. Sloane, Jul 19 2017

			Row 6 is 1 10 27 29 12 1 generating 1 11 48 141 ... (A008780) the seventh term in the Euler transforms of 1,1,1,...; 1,2,3,...; 1,3,6,... 1,4,10,... etc.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 6, 11, 7, 1;
1, 10, 27, 29, 12, 1;
1, 14, 57, 96, 72, 21, 1;
1, 21, 117, 277, 319, 176, 38, 1;
...

N. J. A. Sloane, Transforms

Cf. A000293, A116673 (row sums), A008778 - A008780, A289656.

cp's OEIS Frontend

A131056 A007318 * A131055.

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A132047 3*A007318 - 2*A103451 as infinite lower triangular matrices.

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A152845 Triangle read by rows, A007318 rows repeated seven times .

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A175807 A007318-perfect numbers.

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A175811 A007318-deficient numbers.

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A055375 Euler transform of Pascal's triangle A007318.

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A073617 Consider Pascal's triangle A007318; a(n) = product of terms at +45 degrees slope with the horizontal.

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A132047 3A007318 - 2A103451 as infinite lower triangular matrices.