A059570 -id:A059570 - cp's OEIS frontend

A091597 Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).

1, 1, 1, 3, 2, 1, 5, 5, 3, 1, 11, 10, 8, 4, 1, 21, 21, 18, 12, 5, 1, 43, 42, 39, 30, 17, 6, 1, 85, 85, 81, 69, 47, 23, 7, 1, 171, 170, 166, 150, 116, 70, 30, 8, 1, 341, 341, 336, 316, 266, 186, 100, 38, 9, 1, 683, 682, 677, 652, 582, 452, 286, 138, 47, 10, 1

Offset: 0

Paul Barry, Jan 23 2004

A Jacobsthal-Pascal triangle.

Equals triangle M * Pascal's triangle, M = an infinite lower triangular Toeplitz matrix with A078008: [1, 0, 2, 2, 6, 10, 22, 42, ...] in every column. - Gary W. Adamson, May 25 2009

			Triangle begins as:
    1;
    1,   1;
    3,   2,   1;
    5,   5,   3,   1;
   11,  10,   8,   4,   1;
   21,  21,  18,  12,   5,   1;
   43,  42,  39,  30,  17,   6,   1;
   85,  85,  81,  69,  47,  23,   7,  1;
  171, 170, 166, 150, 116,  70,  30,  8, 1;
  341, 341, 336, 316, 266, 186, 100, 38, 9, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Columns include A001045, A000975, A011377.
Row sums are A059570.
Cf. A078008. - Gary W. Adamson, May 25 2009

Flat(List([0..12], n->List([0..n], k-> Sum([0..n], j-> 2^j*Binomial(n-j, k+j)) ))); # G. C. Greubel, Jun 04 2019

[[(&+[2^j*Binomial(n-j, k+j): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 04 2019

A091597 := proc(n,k)
    if k = 0 then
        A001045(n+1) ;
    elif k = n then
        1 ;
    elif k <0 or k > n then
        0 ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Oct 05 2012

Table[Sum[Binomial[n-j, k+j]*2^j, {j,0,n}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2019 *)

{T(n,k) = sum(j=0, n, 2^j*binomial(n-j, k+j))}; \\ G. C. Greubel, Jun 04 2019

[[sum(2^j*binomial(n-j, k+j) for j in (0..n)) for k in (0..n)] for n in [0..12]] # G. C. Greubel, Jun 04 2019

Number triangle: T(n, k) = Sum_{j=0..n} binomial(n-j, k+j)2^j.
Riordan array: (1/(1-x-2*x^2), x/(1-x)).
k-th column has g.f. (1/(1-x-2*x^2))*(x/(1-x))^k.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - 2*T(n-3,k) - 2*T(n-3,k-1), T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=3, T(2,1)=2, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014

A128255 A114219(signed) * A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 3, 10, 13, 7, 1, 3, 15, 27, 23, 9, 1, 4, 21, 48, 57, 36, 1, 4, 28, 78, 118, 104, 52, 13, 1, 5, 36, 118, 218, 246, 172, 71, 15, 1, 5, 45, 170, 370, 510, 458, 265, 93, 17, 1

Offset: 1

Gary W. Adamson, Feb 20 2007

Row sums = A059570: (1, 2, 6, 14, 34, 78, 178,...).

			First few rows of the triangle are:
1;
1, 1;
2, 3, 1;
2, 6, 5, 1;
3, 10, 13, 7, 1;
3, 15, 27, 23, 9, 1;
4, 21, 48, 57, 36, 11, 1;
...

Cf. A103450, A114219, A007318, A059570, A097807, A128229.

Let the signed version of A114219 {1; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,-3,4;...} = M; and P = Pascal's triangle, A007318. Then A128255 = A114219(signed) * A007318.

A284942 Expansion of Sum_{k>=1} mu(k)^2x^k(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 8, 19, 46, 107, 244, 547, 1213, 2665, 5807, 12567, 27042, 57899, 123428, 262115, 554750, 1170538, 2463154, 5170462, 10829234, 22635087, 47223412, 98353299, 204519549, 424665001, 880581806, 1823667221, 3772341661, 7794697759, 16089424392, 33178906531, 68357928558

Offset: 1

Ilya Gutkovskiy, Apr 06 2017

nonn

Total number of squarefree parts in all compositions (ordered partitions) of n.

			a(4) = 19 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1] and 0 + 2 + 2 + 3 + 2 + 3 + 3 + 4 = 19.

Alois P. Heinz, Table of n, a(n) for n = 1..3312
Index entries for sequences related to compositions

Cf. A005117, A008683, A011782, A059570, A097941, A097979, A102291, A281573, A284943.

a:= proc(n) option remember; add(`if`(numtheory[
      issqrfree](j), ceil(2^(n-j-1)), 0)+a(n-j), j=1..n)
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

nmax = 33; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k (1 - x)^2/(1 - 2 x)^2, {k, 1, nmax}], {x, 0, nmax}], x]]

x='x+O('x^34); Vec(sum(k=1, 34, moebius(k) ^2*x^k*(1 - x)^2/(1 - 2*x)^2)) \\ Indranil Ghosh, Apr 06 2017

G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2.

A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)(1 - x)^2/(1 - 2x)^2.

Original entry on oeis.org

0, 1, 3, 8, 20, 47, 110, 251, 564, 1251, 2750, 5994, 12978, 27934, 59825, 127565, 270959, 573575, 1210466, 2547562, 5348385, 11203292, 23419629, 48865346, 101782870, 211670094, 439548898, 911515214, 1887865266, 3905400206, 8070139762, 16658958223, 34355273843

Offset: 1

Ilya Gutkovskiy, Apr 06 2017

nonn

Total number of prime power parts (1 excluded) in all compositions (ordered partitions) of n.

			a(5) = 20 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 4], [1, 3, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 3], [1, 1, 2, 1], [1, 1, 1, 2], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 0 = 20.

Alois P. Heinz, Table of n, a(n) for n = 1..3313
Index entries for sequences related to compositions

Cf. A011782, A059570, A073335, A097941, A097979, A102291, A246655, A284942.

b:= proc(n) option remember; nops(ifactors(n)[2])=1 end:
a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(b(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Aug 07 2019

nmax = 33; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[k]] x^k (1 - x)^2/(1 - 2 x)^2, {k, 2, nmax}], {x, 0, nmax}], x]]

x='x+O('x^34); concat([0], Vec(sum(k=2, 34, (1\omega(k))*x^k*(1 - x)^2/(1 - 2*x)^2))) \\ Indranil Ghosh, Apr 06 2017

G.f.: Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.

A102289 Total number of odd lists in all sets of lists, cf. A000262.

Original entry on oeis.org

0, 1, 2, 15, 76, 665, 5286, 56287, 597080, 7601841, 99702730, 1484554511, 23049638052, 393702612745, 7036703742446, 135702811542495, 2737989749177776, 58848546456947297, 1321063959370833810, 31310238786268648591, 773291778432688011260, 20031956775840631151481

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A052852, A066897, A066898, A059570, A059570, A081358, A092691.

G:=(x/(1-x^2))*exp(x/(1-x)): Gser:=series(G,x=0,25): seq(n!*coeff(Gser,x^n),n=1..22); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::odd, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Rest[CoefficientList[Series[x/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
nxt[{n_,a_,b_,c_}]:={n+1,b,c,(n+1)*c+(n+1)^2*b-(n-1)^2 (n+1)*a}; NestList[ nxt,{2,0,1,2},30][[All,2]] (* Harvey P. Dale, Jan 13 2019 *)

E.g.f.: x/(1-x^2)*exp(x/(1-x)).
a(n) = n*a(n-1) + n^2*a(n-2) - (n-2)^2*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 + 7/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013

More terms from Emeric Deutsch, Jun 24 2005

a(0)=0 pepended by Alois P. Heinz, May 10 2016

A102290 Total number of even lists in all sets of lists, cf. A000262.

Original entry on oeis.org

0, 0, 2, 6, 60, 380, 3990, 37002, 450296, 5373720, 76018410, 1096730030, 17814654132, 299645294676, 5511836578430, 105550556136690, 2171244984679920, 46545825736022192, 1059273836225051346, 25100215228045842390, 626204775725372971820, 16239127347086448236460

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A052852, A059570, A066897, A066898, A081358, A092691.

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[0,0]cat[Factorial(n)*(&+[(-1)^(n+j)*l(j,-1): j in [0..n-2]]): n in [2..30]]; // G. C. Greubel, Mar 09 2021

Gser:=series(x^2*exp(x/(1-x))/(1-x^2),x=0,22):seq(n!*coeff(Gser,x^n),n=1..21); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::even, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Rest[CoefficientList[Series[x^2/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
Table[If[n<2, 0, n!*Sum[(-1)^(n-j)*LaguerreL[j, -1], {j,0,n-2}]], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[0,0]+[factorial(n)*sum((-1)^(n+j)*gen_laguerre(j,0,-1) for j in (0..n-2)) for n in (2..30)] # G. C. Greubel, Mar 09 2021

E.g.f.: x^2/(1-x^2)*exp(x/(1-x)).
Recurrence: (n-2)*a(n) = (n-2)*n*a(n-1) + (n-1)^2*n*a(n-2) - (n-3)*(n-2)*(n-1)*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 - 41/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = n! * Sum_{j=0..n-2} (-1)^(n+j)*LaguerreL(j, -1) for n>1 with a(0)=a(1)=0. - G. C. Greubel, Mar 09 2021

More terms from Emeric Deutsch, Mar 27 2005

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A275434 Sum of the degrees of asymmetry of all compositions of n.

Original entry on oeis.org

0, 0, 0, 2, 4, 12, 28, 68, 156, 356, 796, 1764, 3868, 8420, 18204, 39140, 83740, 178404, 378652, 800996, 1689372, 3553508, 7456540, 15612132, 32622364, 68040932, 141674268, 294533348, 611436316, 1267611876, 2624702236, 5428361444, 11214636828

Offset: 0

Emeric Deutsch, Jul 29 2016

The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).

A sequence is palindromic if and only if its degree of asymmetry is 0.

			a(4) = 4 because the compositions 4, 13, 22, 31, 112, 121, 211, 1111 have degrees of asymmetry 0, 1, 0, 1, 1, 0, 1, 0, respectively.

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A059570, A275433.

g := 2*z^3*(1-z)/((1-2*z)*(1-z-2*z^2)): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);
a := proc(n) if n = 0 then 0 elif n = 1 then 0 else -(4/9)*(-1)^n+(1/36)*(3*n-2)*2^n end if end proc: seq(a(n), n = 0 .. 32);

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[b[n - j, If[i==0, j, 0]] If[i > 0 && i != j, x, 1], {j, 1, n}]]];
a[n_] := Function[p, Sum[i Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0]];
a /@ Range[0, 32] (* Jean-François Alcover, Nov 24 2020, after Alois P. Heinz in A275433 *)

G.f.: g(z) = 2*z^3*(1-z)/((1-2*z)*(1-z-2*z^2)). In the more general situation of compositions into a[1]=1} z^(a[j]), we have g(z) = (F(z)^2 - F(z^2))/((1+F(z))*(1-F(z))^2).
a(n) = -(4/9)*(-1)^n + (3*n - 2)*2^n/36 for n>=2; a(0) = a(1) = 0.
a(n) = Sum_{k>=0} k*A275433(n,k).
a(n) = 2*A059570(n-2) for n>=3. - Alois P. Heinz, Jul 29 2016

A366987 Triangle read by rows: T(n, k) = -(2^(n - k)*(-1)^n + 2^k + (-1)^k)/3.

Original entry on oeis.org

-1, 0, 0, -2, -1, -2, 2, 1, -1, -2, -6, -3, -3, -3, -6, 10, 5, 1, -1, -5, -10, -22, -11, -7, -5, -7, -11, -22, 42, 21, 9, 3, -3, -9, -21, -42, -86, -43, -23, -13, -11, -13, -23, -43, -86, 170, 85, 41, 19, 5, -5, -19, -41, -85, -170, -342, -171, -87, -45, -27, -21, -27, -45, -87, -171, -342

Offset: 0

Paul Curtz and Thomas Scheuerle, Oct 31 2023

			Triangle T(n, k) starts:
   -1
    0   0
   -2  -1  -2
    2   1  -1  -2
   -6  -3  -3  -3  -6
   10   5   1  -1  -5 -10
  -22 -11  -7  -5  -7 -11 -22
   42  21   9   3  -3  -9 -21 -42
   ...
Note the symmetrical distribution of the absolute values of the terms in each row.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened)

Rows sums: -A282577(n+2), if the conjectures from Colin Barker in A282577 are true.
First column: -(-1)^n * A078008(n).
Second column: (-1)^n * A001045(n).
Third column: -A140966(n).
Fourth column: (-1)^n * A155980(n+2).
Cf. A000079, A020989, A048573, A053088, A059570, A077898, A140966.

T := (n, k) -> -(2^(n-k)*(-1)^n + 2^k + (-1)^k)/3:
seq(seq(T(n, k), k = 0..n), n = 0..10);  # Peter Luschny, Nov 02 2023

A366987row[n_]:=Table[-(2^(n-k)(-1)^n+2^k+(-1)^k)/3,{k,0,n}];Array[A366987row,15,0] (* Paolo Xausa, Nov 07 2023 *)

T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ Thomas Scheuerle, Nov 01 2023

T(n, 0) = -((-2)^n + 2)/3.
T(n, k+1) - T(n, k) = T(n-1, k) + (-1)^k.
T(2*n+1, n) = A001045(n).
T(2*n+1, n+1) = -A001045(n).
T(2*n, n+1) = -A048573(n-1), for n > 0.
Note that the definition of T extends to negative parameters:
T(2*n-2, n-1) = -A001045(n).
-2^n*Sum_{k=0..n} (-1)^k*T(-n, -k) = A059570(n+1).
-4^n*Sum_{k=0..2*n} T(-2*n, -k) = A020989(n).
-Sum_{k=0..n} (-1)^k*T(n, k) = A077898(n). See also A053088.
Sum_{k = 0..2*n} |T(2*n, k)| = (4^(n+1) - 1)/3.
Sum_{k = 0..2*n+1} |T(2*n+1, k)| = (1 + (-1)^n - 2^(2 + n) + 2^(1 + 2*n))/3.
G.f.: (-1 - x + x*y)/((1 - x)*(1 + 2*x)*(1 + x*y)*(1 - 2*x*y)). - Stefano Spezia, Nov 03 2023

a(42) corrected by Paolo Xausa, Nov 07 2023

cp's OEIS Frontend

A091597 Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).

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A128255 A114219(signed) * A007318.

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A284942 Expansion of Sum_{k>=1} mu(k)^2*x^k*(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

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A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.

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A102289 Total number of odd lists in all sets of lists, cf. A000262.

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A102290 Total number of even lists in all sets of lists, cf. A000262.

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Magma

Maple

Mathematica

Sage

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Extensions

A275434 Sum of the degrees of asymmetry of all compositions of n.

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A284942 Expansion of Sum_{k>=1} mu(k)^2x^k(1 - x)^2/(1 - 2*x)^2, where mu() is the Moebius function (A008683).

A284943 Expansion of Sum_{p prime, k>=1} x^(p^k)(1 - x)^2/(1 - 2x)^2.