A028310 -id:A028310 - cp's OEIS frontend

A335545 A(n,k) is the sum of the k-th powers of the (positive) number of permutations of [n] with j descents (j=0..max(0,n-1)); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 1, 2, 18, 24, 5, 1, 1, 2, 66, 244, 120, 6, 1, 1, 2, 258, 2664, 5710, 720, 7, 1, 1, 2, 1026, 29284, 322650, 188908, 5040, 8, 1, 1, 2, 4098, 322104, 19888690, 55457604, 8702820, 40320, 9, 1, 1, 2, 16386, 3543124, 1276095330, 16657451236, 17484605040, 524888040, 362880, 10

Offset: 0

Alois P. Heinz, Sep 12 2020

			Square array A(n,k) begins:
  1,   1,      1,        1,           1,             1, ...
  1,   1,      1,        1,           1,             1, ...
  2,   2,      2,        2,           2,             2, ...
  3,   6,     18,       66,         258,          1026, ...
  4,  24,    244,     2664,       29284,        322104, ...
  5, 120,   5710,   322650,    19888690,    1276095330, ...
  6, 720, 188908, 55457604, 16657451236, 5025377832180, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..60, flattened

Columns k=0-2 give: A028310, A000142, A168562.
Rows n=0+1, 2-3 give: A000012, A007395(k+1), A178789(k+1).
Main diagonal gives A335546.
Cf. A008292, A173018, A334622.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      expand(add(b(u-j, o+j-1, 1)*x^t, j=1..u))+
             add(b(u+j-1, o-j, 1), j=1..o))
    end:
A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= (n, k)-> add(combinat[eulerian1](n, j)^k, j=0..max(0, n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

B[n_, k_] := B[n, k] = Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
A[0, ] = 1; A[n, k_] := Sum[B[n, j]^k, {j, 0, n-1}];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

A(n,k) = Sum_{j=0..max(0,n-1)} A173018(n,j)^k.

A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 12, 14, 14, 15, 15, 18, 17, 19, 18, 20, 20, 22, 21, 25, 23, 26, 25, 28, 26, 29, 27, 31, 29, 32, 31, 34, 33, 35, 34, 38, 35, 41, 37, 42, 40, 43, 41, 45, 42, 46, 44, 48, 45, 50, 46, 52, 49, 53

Offset: 0

Gus Wiseman, Oct 13 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			Representative partitions for each of the a(16) = 11 alternating products:
         (16) -> 16
     (14,1,1) -> 14
     (12,2,2) -> 12
     (10,3,3) -> 10
      (8,4,4) -> 8
  (9,3,2,1,1) -> 6
     (10,4,2) -> 5
     (12,3,1) -> 4
  (6,4,2,2,2) -> 3
     (10,5,1) -> 2
        (8,8) -> 1

The even-length version is A000035.
The non-reverse version is A028310.
The version for factorizations has special cases:
- no changes: A046951
- non-reverse: A046951
- non-integer: A038548
- odd-length: A046951 + A010052
- non-reverse non-integer: A347460
- non-integer odd-length: A347708
- non-reverse odd-length: A046951 + A010052
- non-reverse non-integer odd-length: A347708
The odd-length version is a(n) - A059841(n).
These partitions are counted by A347445, non-reverse A347446.
Counting non-integers gives A347462, non-reverse A347461.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A119620 counts partitions with alternating product 1, ranked by A028982.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A304792 counts distinct subset-sums of partitions.
A325534 counts separable partitions, complement A325535.
A345926 counts possible alternating sums of permutations of prime indices.
Cf. A000070, A002033, A002219, A108917, A122768, A325765, A344654, A344740, A347443, A347444, A347448, A347449.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[Union[revaltprod/@IntegerPartitions[n]],IntegerQ]],{n,0,30}]

A000660 Boustrophedon transform of 1,1,2,3,4,5,...

Original entry on oeis.org

1, 2, 5, 14, 41, 136, 523, 2330, 11857, 67912, 432291, 3027166, 23125673, 191389108, 1705788659, 16289080922, 165919213089, 1795666675824, 20576824369027, 248892651678198, 3168999664907705, 42366404751871660, 593368400878431795, 8688251294851280594

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A000111, A028310, A109449, A231179.

a000660 n = sum $ zipWith (*) (a109449_row n) (1 : [1..])
-- Reinhard Zumkeller, Nov 04 2013

seq(coeff(series(factorial(n)*(x*exp(x)+1)*(sec(x)+tan(x)), x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 30 2018

a[n_] := n! SeriesCoefficient[(1+x Exp[x])(1+Sin[x])/Cos[x], {x, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 30 2018, after Sergei N. Gladkovskii *)

from itertools import accumulate, count, islice
def A000660_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A000660_list = list(islice(A000660_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Algorithm of L. Seidel (1877)
def A000660_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in range(n) :
        Am = i
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        print([A[z] for z in (-i//2..i//2)])
        R.append(A[e*i//2])
    return R
A000660_list(10) # Peter Luschny, Jun 02 2012

a(n) = Sum_{k=0..n} A109449(n,k)*A028310(k). - Reinhard Zumkeller, Nov 04 2013
E.g.f.: (x*exp(x) + 1)*(sec(x) + tan(x)). - Sergei N. Gladkovskii, Oct 28 2014
a(n) = A231179(n) + A000111(n). - Sergei N. Gladkovskii, Oct 28 2014
a(n) ~ n! * (2 + Pi*exp(Pi/2)) * (2/Pi)^(n+1). - Vaclav Kotesovec, Jun 12 2015

A078444 Floor of geometric mean of two consecutive primes.

Original entry on oeis.org

2, 3, 5, 8, 11, 14, 17, 20, 25, 29, 33, 38, 41, 44, 49, 55, 59, 63, 68, 71, 75, 80, 85, 92, 98, 101, 104, 107, 110, 119, 128, 133, 137, 143, 149, 153, 159, 164, 169, 175, 179, 185, 191, 194, 197, 204, 216, 224, 227, 230, 235, 239, 245, 253, 259, 265, 269, 273, 278

Offset: 1

Lior Manor, Dec 31 2002

For n > 1, a(n) = prime(n) iff prime(n) and prime(n+1) are twin primes.

			a(7) = floor(sqrt(prime(7)*prime(8))) = 17.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Andrica's Conjecture

Cf. A001223, A006094.

[Floor(Sqrt(NthPrime(n)*NthPrime(n+1))): n in [1..60]]; // Vincenzo Librandi, Dec 12 2015

seq(floor(sqrt(ithprime(i)*ithprime(i+1))), i=1..100); # Robert Israel, Dec 12 2015

Table[Floor[Sqrt[Prime[n] Prime[n + 1]]], {n, 60}] (* Vincenzo Librandi, Dec 12 2015 *)
Table[Ceiling[(Prime[n] + Prime[n + 1])/2 - 1], {n, 100}] (* Miko Labalan, Dec 14 2015 *)

a(n) = sqrtint(prime(n)*prime(n+1)); \\ Michel Marcus, Dec 12 2015

a(n) = floor(sqrt(prime(n)*prime(n+1))).
From Miko Labalan, Dec 12 2015: (Start)
a(n) = A006254(A028310(n - 1)) + A067076(n);
a(n) = A067076(A028310(n - 1)) + A006254(n);
a(n) = A005097(A028310(n - 1)) + A005097(n).
(End)
For n >= 2 these formulas are equivalent to sqrt(prime(n)*prime(n+1)) > (prime(n)+prime(n+1))/2 - 1, and thus to A001223(n) <= 2 + 2*sqrt(2*prime(n)). This would be implied by Andrica's conjecture, but is as yet unproven. - Robert Israel, Dec 13 2015

A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u(u+1)...*(u+n-1) du.

Original entry on oeis.org

2, 0, 4, 8, 232, 448, 18224, 35424, 1036064, 2025472, 130960832, 257072000, 689908475264, 1358275350528, 8031885897472, 15847920983552, 7981032500085248, 15774370258485248, 12448755354530366464

Offset: 0

Paul Curtz, Sep 20 2011

nonn

Numerators of the second row of an array based on Adams numerical integration. Take q!*s(m,q) = Integral_{-m-1..1} u*(u+1)*...*(u+q-1) du. a(n) is in the second row (case m=0) numerators of s(m,q) in the comments.
The unreduced array s(m,q), (m=-1,0,1,...,   columns q=0,1,2,...) is
1,   1/2,   5/12,   9/24,   251/720, 475/1440,  = A002657(n)/A091137(n),
2,     0,   4/12,   8/24,   232/720, 448/1440,  = a(n)/A091137(n),
3,  -3/2,   9/12,   9/24,   243/720, 459/1440,
4,  -8/2,  32/12,      0,   224/720, 448/1440,
5, -15/2,  85/12, -55/24,   475/720, 475/1440,
6, -24/2, 180/12, -216/24, 2376/720,        0.
Column numerators: A000027, -A067998(n), A152064(n), A157371(n), A165281(n).
Page 56 of the reference.
(*) 2/2 = 1,
    2/2 + 0 = 1,
    2/3 + 0 + 1/3 = 1,
    2/4 + 0 + 1/6 + 1/3 = 1. Reduced.

P. Curtz, Intégration numérique des systèmes differentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.

A195287 := proc(n)
        mul(u+i,i=0..n-1) ;
        int(%,u=-1..1) ;
        %/n!*A091137(n) ;
end proc:
seq(A195287(n),n=0..20) ; # R. J. Mathar, Oct 02 2011

(* a7 = A091137 *) a7[n_] := a7[n] = Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*a7[n-1]; a7[0]=1; a[n_] := a7[n]/n!*Integrate[ Pochhammer[u, n], {u, -1, 1}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 13 2012 *)

b(n) = a(n)/A091137(n).
b(0)/2 = 1,
b(0)/2 + b(1) = 1,
b(0)/3 + b(1)/2 + b(2) = 1,
b(0)/4 + b(1)/3 + b(2)/2 + b(3) = 1.
First vertical denominators: A028310(n) + 1. See A104661.
Values in (*).

A212393 Expansion of (1-4x+7x^2-5x^3+4x^4-6x^5+21x^6+18x^7-5x^8)/(1-x)^5.

Original entry on oeis.org

1, 1, 2, 5, 14, 30, 72, 195, 485, 1059, 2065, 3682, 6120, 9620, 14454, 20925, 29367, 40145, 53655, 70324, 90610, 115002, 144020, 178215, 218169, 264495, 317837, 378870, 448300, 526864, 615330, 714497, 825195, 948285, 1084659, 1235240, 1400982, 1582870, 1781920

Offset: 0

Bruno Berselli, May 14 2012

In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(0,k,0,0)(x,0) represents a generating function depending on k and x.
For successive values of k we have:
k=1, the g.f. of A000012: 1/(1-x);
k=2, the g.f. of A028310: (1-x+x^2)/(1-x)^2;
k=3, the g.f. (1-2*x+2*x^2+x^3-x^4)/(1-x)^3, whose coefficients (except the first two) are given by A000096 (for n>0);
k=4, the g.f. (1-3*x+4*x^2-x^3+3*x^4-5*x^5+2*x^6)/(1-x)^4, whose coefficients (except the first three) are given by A005586 (for n>0).
This sequence corresponds to the case k=5.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (page 21, Theorem 10).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

m:=39; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5));

CoefficientList[Series[(1 - 4 x + 7 x^2 - 5 x^3 + 4 x^4 - 6 x^5 + 21 x^6 + 18 x^7 - 5 x^8)/(1 - x)^5, {x, 0, 38}], x]

Vec((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5+O(x^39))

G.f.: (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>8, a(0)=a(1)=1, a(2)=2, a(3)=5, a(4)=14, a(5)=30, a(6)=72, a(7)=195, a(8)=485.
a(n) = (n-3)*(31*n^3-369*n^2+1454*n-1560)/24 for n>3, a(0)=a(1)=1, a(2)=2, a(3)=5.
G.f.: 1+x+2*x^2+5*x^3 + 14*x^4*G(0), where G(k)= 1 + x*(k+1)*(124*k^3+192*k^2+89*k+180)/( (2*k+1)*(62*k^3+3*k^2-5*k+84) - x*(2*k+1)*(62*k^3+3*k^2-5*k+84)*(2*k+3)*(62*k^3+189*k^2+187*k+144)/(x*(2*k+3)*(62*k^3+189*k^2+187*k+144) + (k+1)*(124*k^3+192*k^2+89*k+180)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013

A215796 Number of distinct values taken by 7th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 283, 691, 1681, 3988, 9241, 20681, 44217, 89644

Offset: 1

Alois P. Heinz, Aug 24 2012

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 7th derivative at x=1: (x^(x^(x^x))) -> 26054; ((x^x)^(x^x)), ((x^(x^x))^x) -> 41090; (x^((x^x)^x)) -> 47110; (((x^x)^x)^x) -> 70098.

Column k=7 of A216368.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215837.

T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
      seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
      combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
    end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
      nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
    end():
a:= n-> nops({map(f-> 7!*coeff(series(subs(x=x+1, f), x, 8), x, 7), T(n))[]}):
seq(a(n), n=1..12);

A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 1, 3, 4, 5, 1, 2, 4, 5, 6, 1, 2, 4, 5, 6, 7, 1, 2, 3, 5, 6, 7, 8, 1, 2, 3, 5, 6, 7, 8, 9, 1, 2, 3, 4, 6, 7, 8, 9, 10, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13

Offset: 1

Antti Karttunen, Feb 10 2014

This is transpose of A237447, please see comments there.

			The top left 9 X 9 corner of this infinite square array:
  1 2 3 4 5 6 7 8 9
  2 1 1 1 1 1 1 1 1
  3 3 2 2 2 2 2 2 2
  4 4 4 3 3 3 3 3 3
  5 5 5 5 4 4 4 4 4
  6 6 6 6 6 5 5 5 5
  7 7 7 7 7 7 6 6 6
  8 8 8 8 8 8 8 7 7
  9 9 9 9 9 9 9 9 8

Antti Karttunen, Table of first 144 antidiagonals of array, flattened

Transpose: A237447.
The leftmost column and the topmost row: A000027. Second row: A054977. Central diagonal: A028310 (note the different starting offsets).
Antidiagonal sums: A074148.
Cf. A237265, A000330, A002024, A002260, A004736, A000124, A010054.

(define (A237448 n) (cond ((= 1 (A010054 (- n 1))) (A002024 n)) ((< (A004736 n) (A002260 n)) (A002260 n)) (else (- (A002260 n) 1))))

As a one-dimensional sequence:
If A010054(n-1) = 1 [that is, if n is in A000124], then a(n) = A002024(n), otherwise, if A004736(n) < A002260(n), a(n) = A002260(n), and if A004736(n) >= A002260(n), a(n) = A002260(n)-1.
Equivalently, as a square array T:
When col < row, T(row,col) = row, for 1 < row <= col, T(row,col) = row-1, and for the first row T(1,col) = col = A000027(col).
Can be computed also as a transposed version of the infinite limit of the finite square arrays in sequence A237265: T(row,col) = A237265((A000330(max(row,col)-1)+1) + (max(row,col)*(col-1)) + (row-1)).

A255264 Total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562 after A048645(n) generations.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 85, 89, 101, 149, 341, 345, 357, 405, 597, 1365, 1369, 1381, 1429, 1621, 2389, 5461, 5465, 5477, 5525, 5717, 6485, 9557, 21845, 21849, 21861, 21909, 22101, 22869, 25941, 38229, 87381, 87385, 87397, 87445, 87637

Offset: 1

Omar E. Pol, Feb 19 2015

It appears that these are the terms of A147562, A162795, A169707, A255366, A256250, A256260, whose indices have binary weight 1 or 2.

			Also, written as an irregular triangle in which row lengths are the terms of A028310 the sequence begins:
      1;
      5;
      9,    21;
     25,    37,    85;
     89,   101,   149,   341;
    345,   357,   405,   597,  1365;
   1369,  1381,  1429,  1621,  2389,  5461;
   5465,  5477,  5525,  5717,  6485,  9557, 21845;
  21849, 21861, 21909, 22101, 22869, 25941, 38229, 87381;
  ...
Right border gives the positive terms of A002450.
It appears that the second leading diagonal gives the odd terms of A206374.

Cf. A002450, A028310, A032925, A048645, A075897, A139250, A147562, A162795, A169707, A206374, A209492, A255263, A255366, A256250, A256260.

a(n) = A147562(A048645(n)).
Conjecture 1: a(n) = A162795(A048645(n)).
Conjecture 2: a(n) = A169707(A048645(n)).
Conjecture 3: a(n) = A255366(A048645(n)).
Conjecture 4: a(n) = A256250(A048645(n)).
Conjecture 5: a(n) = A256260(A048645(n)).
a(n) = A032925(A209492(n-1)) (conjectured). - Jon Maiga, Dec 17 2021

A328902 Triangle T(n, k) read by rows: T(n, k) is the denominator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for 0 <= k <= n, n > 0; T(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 2, 1, 4, 5, 1, 1, 1, 1, 5, 6, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 8, 1, 2, 1, 4, 1, 2, 1, 8, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 10, 1, 2, 1, 2, 5, 2, 1, 1, 1, 5, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 12, 1, 2, 3, 4, 1, 3, 1, 2, 3, 1, 1, 6

Offset: 0

Stefano Spezia, Oct 30 2019

			n\k| 0 1 2 3 4 5 6
---+--------------
0  | 1
1  | 1 1
2  | 2 1 2
3  | 3 1 1 3
4  | 4 1 2 1 4
5  | 5 1 1 1 1 5
6  | 6 1 2 3 1 1 1
...

Stefano Spezia, First 141 rows of the triangle, flattened
D. Armstrong, N. A. Loehr, G. S. Warrington, Rational Parking Functions and Catalan Numbers, Annals of Combinatorics (2016), Volume 20, Issue 1, pp 21-58.
M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.

Cf. A000108, A028310 (1st column), A046899, A051162, A328901 (numerator).

Flatten[Join[{1},Table[(n+k)/GCD[n+k,Binomial[n+k,n]],{n,1,12},{k,0,n}]]]

A328902(n,k)=if(n,(n+k)/gcd(binomial(n+k,n),n+k),1) \\ M. F. Hasler, Nov 04 2019

T(n, k) = (n + k)/gcd(binomial(n + k, n), n + k) for n > 0.

cp's OEIS Frontend

A335545 A(n,k) is the sum of the k-th powers of the (positive) number of permutations of [n] with j descents (j=0..max(0,n-1)); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

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A000660 Boustrophedon transform of 1,1,2,3,4,5,...

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A078444 Floor of geometric mean of two consecutive primes.

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A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u*(u+1)*...*(u+n-1) du.

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A212393 Expansion of (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.

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A215796 Number of distinct values taken by 7th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

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A195287 a(n) = (A091137(n)/n!) * Integral_{u=-1..1} u(u+1)...*(u+n-1) du.

A212393 Expansion of (1-4x+7x^2-5x^3+4x^4-6x^5+21x^6+18x^7-5x^8)/(1-x)^5.