A091044 -id:A091044 - cp's OEIS frontend

A034870 Even-numbered rows of Pascal's triangle.

1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 15, 20, 15, 6, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91, 14, 1

Offset: 0

N. J. A. Sloane

The sequence of row lengths of this array is [1,3,5,7,9,11,13,...]= A005408(n), n>=0.
Equals X^n * [1,0,0,0,...] where X = an infinite tridiagonal matrix with (1,1,1,...) in the main and subsubdiagonal and (2,2,2,...) in the main diagonal. X also = a triangular matrix with (1,2,1,0,0,0,...) in each column. - Gary W. Adamson, May 26 2008
a(n,m) has the following interesting combinatoric interpretation. Let s(n,m) equal the set of all base-4, n-digit numbers with n-m more 1-digits than 2-digits. For example s(2,1) = {10,01,13,31} (note that numbers like 1 are left-padded with 0's to ensure that they have 2 digits). Notice that #s(2,1) = a(2,1) with # indicating cardinality. This is true in general. a(n,m)=#s(n,m). In words, a(n,m) gives the number of n-digit, base-4 numbers with n-m more 1 digits than 2 digits. A proof is provided in the Links section. - Russell Jay Hendel, Jun 23 2015

			Triangle begins:
  1;
  1,  2,  1;
  1,  4,  6,   4,   1;
  1,  6, 15,  20,  15,   6,   1;
  1,  8, 28,  56,  70,  56,  28,   8,   1;
  1, 10, 45, 120, 210, 252, 210, 120,  45,  10,  1;
  1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1;

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Peter Bala, Notes on generalized Riordan arrays
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
Russell Jay Hendel, Proof that a(n,m) gives the number of n-digit, base-4 numbers with n-m more 1-digits than 2-digits.
Wolfdieter Lang, First 9 rows.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A034871.
Cf. A000302 (row sums, powers of 4), alternating row sums are 0, except for n=0 which gives 1.
Cf. A003095, A034839, A080928, A086645.

a034870 n k = a034870_tabf !! n !! k
a034870_row n = a034870_tabf !! n
a034870_tabf = map a007318_row [0, 2 ..]
-- Reinhard Zumkeller, Apr 19 2012, Apr 02 2011

/* As triangle: */ [[Binomial(n,k): k in [0..n]]: n in [0.. 15 by 2]]; // Vincenzo Librandi, Jul 16 2015

T := (n,k) -> simplify(GegenbauerC(`if`(kPeter Luschny, May 08 2016

Flatten[Table[Binomial[n,k],{n,0,20,2},{k,0,n}]] (* Harvey P. Dale, Dec 15 2014 *)

taylor(1/(1-x*(y+1)^2),x,0,10,y,0,10); /* Vladimir Kruchinin, Nov 22 2020 */

flatten([[binomial(2*n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

T(n, m) = binomial(2*n, m), 0<= m <= 2*n, 0<=n, else 0.
G.f. for column m=2*k sequence: (x^k)*Pe(k, x)/(1-x)^(2*k+1), k>=0; for column m=2*k-1 sequence (x^k)*Po(k, x)/(1-x)^(2*k), k>=1, with the row polynomials Pe(k, x) := sum(A091042(k, m)*x^m, m=0..k) and Po(k, x) := 2*sum(A091044(k, m)*x^m, m=0..k-1); see also triangle A091043.
From Paul D. Hanna, Apr 18 2012: (Start)
Let A(x) be the g.f. of the flattened sequence, then:
G.f.: A(x) = Sum_{n>=0} x^(n^2) * (1+x)^(2*n).
G.f.: A(x) = Sum_{n>=0} x^n*(1+x)^(2*n) * Product_{k=1..n} (1 - (1+x)^2*x^(4*k-3)) / (1 - (1+x)^2*x^(4*k-1)).
G.f.: A(x) = 1/(1 - x*(1+x)^2/(1 + x*(1-x^2)*(1+x)^2/(1 - x^5*(1+x)^2/(1 + x^3*(1-x^4)*(1+x)^2/(1 - x^9*(1+x)^2/(1 + x^5*(1-x^6)*(1+x)^2/(1 - x^13*(1+x)^2/(1 + x^7*(1-x^8)*(1+x)^2/(1 - ...))))))))), a continued fraction.
(End)
From Peter Bala, Jul 14 2015: (Start)
Denote this array by P. Then P * transpose(P) is the square array ( binomial(2*n + 2*k, 2*k) )n,k>=0, which, read by antidiagonals, is A086645.
Transpose(P) is a generalized Riordan array (1, (1 + x)^2) as defined in the Bala link.
Let p(x) = (1 + x)^2. P^2 gives the coefficients in the expansion of the polynomials ( p(p(x)) )^n, P^3 gives the coefficients in the expansion of the polynomials ( p(p(p(x))) )^n and so on.
Row sums are 2^(2*n); row sums of P^2 are 5^(2*n), row sums of P^3 are 26^(2*n). In general, the row sums of P^k, k = 0,1,2,..., are equal to A003095(k)^(2*n).
The signed version of this array ( (-1)^k*binomial(2*n,k) )n,k>=0 is a left-inverse for A034839.
A034839 * P = A080928. (End)
T(n, k) = GegenbauerC(m, -n, -1) where m = k if kPeter Luschny, May 08 2016
G.f.: 1/(1-x*(y+1)^2). - Vladimir Kruchinin, Nov 22 2020

A111910 Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 84, 30, 1, 1, 55, 330, 330, 55, 1, 1, 91, 1001, 2145, 1001, 91, 1, 1, 140, 2548, 10010, 10010, 2548, 140, 1, 1, 204, 5712, 37128, 68068, 37128, 5712, 204, 1, 1, 285, 11628, 116280, 352716, 352716, 116280, 11628, 285, 1

Offset: 0

Emeric Deutsch, Aug 19 2005

			Array S(n,k) in rectangular form (n, k >= 0):
  1,  1,    1,     1,    1,       1,       1,       1,        1, ...
  1,  5,   14,    30,   55,      91,     140,     204,      285, ...
  1, 14,   84,   330,  1001,   2548,    5712,   11628,    21945, ...
  1, 30,  330,  2145, 10010,  37128,  116280,  319770,   793155, ...
  1, 55, 1001, 10010, 68068, 352716, 1492260, 5393454, 17185025, ...
  ...
Array T(n,k) in triangular form (n >= 0 and 0 <= k <= n):
  1,
  1,  1,
  1,  5,    1,
  1, 14,   14,   1,
  1, 30,   84,  30,     1,
  1, 55,  330, 330,    55,  1,
  1, 91, 1001, 2145, 1001, 91, 1,
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Germain Kreweras and Heinrich Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin. 2 (1981), 55-60.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 9.
Anthony J. Wood, Richard A. Blythe, and Martin R. Evans, Combinatorial mappings of exclusion processes, arXiv:1908.00942 [cond-mat.stat-mech], 2019.

Cf. A091044, A111911 (main diagonal), A196148 (row sums of triangle).

Cf. A001263, A174158.

T:= func< n,k | Binomial(n+1, k)*Binomial(2*n+1, 2*k)/((k+1)*(2*k+1)) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021

a:=(p,q)->(p+q+1)!*(2*p+2*q+1)!/(p+1)!/(2*p+1)!/(q+1)!/(2*q+1)!: for n from 0 to 10 do seq(a(j,n-j),j=0..n) od; # yields sequence in triangular form

Table[(# + q + 1)! (2 # + 2 q + 1)!/((# + 1)! (2 # + 1)! (q + 1)! (2 q + 1)!) &[r - q], {r, 0, 9}, {q, 0, r}] // Flatten (* Michael De Vlieger, Oct 21 2019 *)
Table[Binomial[n+1, k]*Binomial[2*n+1, 2*k]/((k+1)*(2*k+1)), {n, 0, 12}, {k, 0,
n}]//Flatten (* G. C. Greubel, Feb 12 2021 *)

def A111910(n,k): return binomial(n+1, k)*binomial(2*n+1, 2*k)/((k+1)*(2*k+1))
flatten([[A111910(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 12 2021

S(n,n) = A111911(n).
From Peter Bala, Oct 13 2011: (Start)
Define a(n) = n!*(n+1/2)!*(n+1)!/(1/2)!.
S(n,k) = a(n+k)/(a(n)*a(k)) gives the sequence as a square array while T(n,k) = a(n)/(a(n-k)*a(k)) gives the sequence as a triangle.
S(n-1,k)*S(n,k+1)*S(n+1,k-1) = S(n-1,k+1)*S(n,k-1)*S(n+1,k). Cf. A091044.
(End)
From G. C. Greubel, Feb 12 2021: (Start)
As a number triangle:
T(n, k) = binomial(n+1, k)*binomial(2*n+1, 2*k)/((k+1)*(2*k+1)).
T(n, k) = binomial(2*n+1, 2*k)/((2*k+1)*binomial(n, k)) * A001263(n+1, k+1). (End)
From Peter Bala, Sep 19 2021: (Start)
As a triangle: T(n,k) = a(n)/(a(n-k)*a(k)), where a(n) = Product_{j = 1..n} s(p,j) with s(p,j) = Sum_{j = 1..n} j^p and p = 2. Note, p = 0 gives Pascal's triangle A007318, p = 1 gives the triangle of Narayana numbers A001263 and p = 3 gives the triangle A174158 whose entries are the squares of the Narayana numbers.
Let E(y) = Sum_{n >= 0} y^n/((n+1)!*(2*n+1)!). Then as a triangle this is the generalized Riordan array (E(y), y) as defined in Wang and Wang with respect to the sequence c_n = (n+1)!*(2*n+1)!.  Cf. A001263.
Generating function: E(y)*E(x*y) = 1 + (1 + x)*y/(2!*3!) + (1 + 5*x + x^2)*y^2/(3!*5!) + (1 + 14*x + 14*x^2 + x^3)*y^3/(4!*7!) + ....
The n-th power of this array has a generating function E(y)^n*E(x*y). In particular, the matrix inverse has a generating function E(x*y)/E(y).
exp(y)*E(y) = 1 + 13*y/(2!*3!) + 421*y^2/(3!*5!) + 25368*y^3/(4!*7!) + ... is essentially a generating function for A081442. (End)

Example section edited by Petros Hadjicostas, Sep 03 2019

A381706 Three-dimensional array of the number b(n, k, i) of permutations of k chosen numbers in {1,2,...,n} with i-1 descents, n >= 1, 1 <= k <= n, 1 <= i <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 11, 11, 1, 1, 11, 11, 1, 1, 1, 1, 1, 1, 1, 6, 6, 6, 1, 1, 16, 26, 16, 1, 1, 26, 66, 26, 1, 1, 26, 66, 26, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 1, 1, 22, 37, 37, 22, 1, 1, 42, 137, 137, 42, 1

Offset: 1

Timothy Y. Chow, Mar 04 2025

For n >= 1, 0 <= k <= n-2, and 0 <= i <= n-1, b(n+1, k+1, i+1) is the (2i)th Betti number of the prepermutohedral variety X_k obtained by k iterated blowups of projective space P^n.

			For n = 4 and k = 2, there are 12 permutations of 2 chosen numbers in {1,2,3,4}; the number of descents is defined to be the number of descents in the permutation of the chosen numbers, plus the number of non-chosen numbers greater than the first chosen number. For example, 32 has 2 descents because 3 is greater than 2 and 4 (a non-chosen number) is greater than 3. The four other permutations of 2 chosen numbers with 2 descents are 31, 12, 13, 14.
The sequence is most naturally viewed as a sequence of squares of size 1x1, 2x2, 3x3, 4x4, etc.
  1               [E(1)]
  1  1
  1  1            [E(2)]
  1  1  1
  1  4  1
  1  4  1         [E(3)]
  1  1  1  1
  1  5  5  1
  1 11 11  1
  1 11 11  1      [E(4)]
  1  1  1  1  1
  1  6  6  6  1
  1 16 26 16  1
  1 26 66 26  1
  1 26 66 26  1   [E(5)]
  ...
[E(n)] refers to row n of A008292.

Timothy Y. Chow, Table of n, a(n) for n = 1..385
Timothy Chow et al., A new generalization of Eulerian numbers, MathOverflow, 2025.
J.-L. Lin, The geometry and combinatorics of some Hessenberg varieties related to the permutohedral variety, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024), Research Paper #P3.17.

Cf. A008292, A068424, A091044, A381888.

beta := proc(n, k)
   local b, p, plist, descents, s, i, r, R:
   r := 1..n; R := {`$`(r)};
   b := Array(r, fill=0);
   plist := combinat:-permute(n, k):
   for p in plist do
      descents := 1:
      s := R minus {op(p)}:
      for i in s do
         if i > p[1] then descents := descents + 1 end if:
      end do:
      for i to k-1 do
         if p[i] > p[i+1] then descents := descents + 1 end if:
      end do:
      b[descents] := b[descents] + 1:
   end do:
   return b
end proc:
for n from 1 to 5 do seq(beta(n, k), k = 1..n) end do;
# After Max Alekseyev's PARI program:
b := (n, k, i) -> local p, t, j; add(add(binomial(n - p, t) * binomial(p - 1, n - k - t) * add(binomial(k, j)*(-1)^j*(i - t - j)^(n - t - p) * (i - 1 - t - j)^(k - n - 1 + t + p), j = 0..i-1-t), t = n+1-k-p..i-1), p = 1..n): seq(seq(seq(b(n, k, j), j = 1..n), k = 1..n), n = 1..6);  # Peter Luschny, Mar 15 2025

{ b(n,k,i) = sum(p=1, n, sum(t=n+1-k-p,i-1, my(l=n+1-t-p); binomial(n-p,t) * binomial(p-1,n-k-t) * sum(j=0,i-1-t, binomial(k,j) * (-1)^j * (i-t-j)^(l-1) * (i-1-t-j)^(k-l) )) ); } \\ Max Alekseyev, Mar 14 2025

def beta(n: int, k: int) -> list[int]:
    b = [0]*n
    for p in Permutations(n, k):
        descents = sum(int(not i in p and i > p[0]) for i in (1..n))
        descents += sum(int(p[i-1] > p[i]) for i in (1..k-1))
        b[descents] += 1
    return b
def Trow(n) -> list[int]: return flatten([beta(n, k) for k in (1..n)])
for n in (1..6): print(f"{n}: ", Trow(n))  # Peter Luschny, Mar 11 2025

Explicit formula for b(n, k, i) is given in MO link and PARI code below. - Max Alekseyev, Mar 14 2025
The recurrence b(n,k,i) = b(n,k-1,i) + (n choose k) Sum_{i=d-n+k}^{d-1} b(k-1,k-2,j) was conjectured by Ron Fertig and proved by Alex R. Miller.
b(n, n, k) equals the Eulerian number T(n, k) of A008292.
If n > 1 then b(n, n-1, k) also equals the Eulerian number T(n, k) of A008292.
Sum_{i} b(n, k, i) equals the falling factorial A068424.
Empirically, Sum_{k} b(n, k, i) seems to equal A381888.

A324010 The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for x >= 0, y >= 0 (the unnormalized second moment). The table is read by antidiagonals.

Original entry on oeis.org

1, 4, 4, 9, 26, 9, 16, 92, 92, 16, 25, 240, 474, 240, 25, 36, 520, 1704, 1704, 520, 36, 49, 994, 4879, 8084, 4879, 994, 49, 64, 1736, 11928, 29560, 29560, 11928, 1736, 64, 81, 2832, 25956, 89928, 134450, 89928, 25956, 2832, 81, 100, 4380, 51648, 238440, 498140, 498140, 238440, 51648, 4380, 100

Offset: 0

Günter Rote, Feb 12 2019

			There are two lattice paths from (0,0) to (x,y)=(1,1): P1=(0,0),(1,0),(1,1) and P2=(0,0),(0,1),(1,1), and hence 4 pairs of lattice paths: (P1,P1),(P1,P2),(P2,P1),(P2,P2). The number of common points is 3,2,2,3, respectively, and the sum of the squares of these numbers is 9+4+4+9 = 26 = a(1,1).
Table begins
   1   4    9    16     25 ...
   4  26   92   240    520 ...
   9  92  474  1704   4879 ...
  16 240 1704  8084  29560 ...
  25 520 4879 29560 134450 ...
  ...

Kevin Buchin, Kenny Chiu, Stefan Felsner, Günter Rote, André Schulz, The number of convex polyominoes with given height and width, arXiv:1903.01095 [math.CO], 2019.

See A306687 for the lower triangular half of the same data, read by rows.

See A091044 for the unnormalized first moment (the sum of the number of common points without squaring).

Table[(# + y + 1) Binomial[# + y + 2, # + 1] Binomial[# + y, #] - Binomial[2 # + 2 y + 2, 2 # + 1]/2 &[x - y], {x, 0, 9}, {y, 0, x}] // Flatten (* Michael De Vlieger, Apr 15 2019 *)

a(x,y) = (x+y+1)*binomial(x+y+2,x+1)*binomial(x+y,x)-binomial(2*x+2*y+2,2*x+1)/2;
matrix(10, 10, n, k, a(n-1,k-1)) \\ Michel Marcus, Apr 08 2019

A(x,y) = (x+y+1) * binomial(x+y+2,x+1) * binomial(x+y,x) - binomial(2*x+2*y+2,2*x+1)/2.

cp's OEIS Frontend

A034870 Even-numbered rows of Pascal's triangle.

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A111910 Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).

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A381706 Three-dimensional array of the number b(n, k, i) of permutations of k chosen numbers in {1,2,...,n} with i-1 descents, n >= 1, 1 <= k <= n, 1 <= i <= n.

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A324010 The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for x >= 0, y >= 0 (the unnormalized second moment). The table is read by antidiagonals.

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