A048574 -id:A048574 - cp's OEIS frontend

A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

1, 4, 12, 30, 68, 144, 289, 556, 1034, 1868, 3292, 5678, 9608, 15984, 26188, 42314, 67509, 106460, 166090, 256552, 392628, 595696, 896484, 1338894, 1985298, 2923840, 4278448, 6222518, 8997544, 12938368, 18507297, 26340040, 37307326, 52597320, 73825504, 103180702

Offset: 2

Ilya Gutkovskiy, Jan 30 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A001934, A002448, A014968, A015128, A048574, A063725, A327380.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..37);  # Alois P. Heinz, Feb 10 2021

nmax = 37; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^2/4, {x, 0, nmax}], x] // Drop[#, 2] &
nmax = 37; CoefficientList[Series[(1/4) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
A015128[n_] := Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; a[n_] := (1/4) Sum[A015128[k] A015128[n - k], {k, 1, n - 1}]; Table[a[n], {n, 2, 37}]

G.f.: (1/4) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^2.
a(n) = Sum_{k=0..n} A014968(k) * A014968(n-k).
a(n) = (1/4) * Sum_{k=1..n-1} A015128(k) * A015128(n-k).
a(n) = (A001934(n) - 2 * A015128(n)) / 4 for n > 0.

A023626 Self-convolution of (1, p(1), p(2), ...).

Original entry on oeis.org

1, 4, 10, 22, 43, 80, 137, 222, 343, 508, 737, 1030, 1411, 1888, 2477, 3198, 4059, 5096, 6297, 7702, 9327, 11176, 13301, 15682, 18355, 21344, 24673, 28358, 32411, 36896, 41769, 47082, 52883, 59148, 65937, 73298, 81251, 89776, 98957

Offset: 1

Clark Kimberling

nonn

p(1),p(2),p(3)... are the prime numbers (A000040). The analogous sequence for the partition numbers is A048574.

			G.f. = x + 4*x^2 + 10*x^3 + 22*x^4 + 43*x^5 + 80*x^6 + 137*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A008578, A014342, A048574.

a023626 n = a023626_list !! (n-2)
a023626_list = f a000040_list [1] where
   f (p:ps) rs = (sum $ zipWith (*) rs a008578_list) : f ps (p : rs)
-- Reinhard Zumkeller, Nov 09 2015

z = 100; p = Join[{1}, Prime[Range[z]]];
a[n_] := Sum[p[[i]] p[[n - i + 1]], {i, 1, n}];
Table[a[n], {n, 1, z}]  (* Clark Kimberling, Dec 01 2016 *)
a[ n_] := If[ n < 1, 0, SeriesCoefficient[ (1 + O[x]^n + Sum[ Prime[k] x^k, {k, n - 1}])^2, {x, 0, n - 1}]]; (* Michael Somos, Dec 01 2016 *)
Table[With[{c=Join[{1},Prime[Range[n]]]},ListConvolve[c,c]],{n,0,40}]// Flatten (* Harvey P. Dale, Oct 19 2018 *)

G.f: x*(1+b(x))^2 = (c(x)^2)/x, where b(x) and c(x) are respectively the g.f. of A000040 and A008578. - Mario C. Enriquez, Dec 10 2016

A304789 Number T(n,k) of partitions of 2n whose Ferrers-Young diagram allows exactly k different domino tilings; triangle T(n,k), n>=0, 0<=k<=A304790(n), read by rows.

Original entry on oeis.org

0, 1, 0, 2, 0, 4, 1, 1, 6, 2, 2, 2, 10, 3, 4, 1, 2, 6, 14, 4, 6, 4, 4, 0, 2, 2, 12, 22, 5, 8, 7, 6, 2, 4, 4, 0, 0, 4, 1, 2, 25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2, 46, 44, 7, 12, 17, 14, 8, 8, 8, 0, 4, 12, 5, 6, 0, 8, 2, 0, 8, 4, 0, 4, 0, 0, 0, 2, 2, 0, 0, 4, 1, 2, 0, 0, 2, 0, 1

Offset: 0

Alois P. Heinz, May 18 2018

			T(2,2) = 1: 22.
T(3,0) = 1: 321.
T(3,1) = 6: 111111, 21111, 3111, 411, 51, 6.
T(3,2) = 2: 2211, 42.
T(3,3) = 2: 222, 33.
T(8,36) = 1: 4444.
Triangle T(n,k) begins:
   0,  1;
   0,  2;
   0,  4, 1;
   1,  6, 2,  2;
   2, 10, 3,  4,  1,  2;
   6, 14, 4,  6,  4,  4, 0, 2, 2;
  12, 22, 5,  8,  7,  6, 2, 4, 4, 0, 0, 4, 1, 2;
  25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2;

Alois P. Heinz, Rows n = 0..20, flattened
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Domino
Wikipedia, Domino tiling
Wikipedia, Ferrers diagram
Wikipedia, Mutilated chessboard problem
Wikipedia, Partition (number theory)
Wikipedia, Young tableau, Diagrams
Index entries for sequences related to dominoes

Columns k=0-1 give: A304710, A139582(n) = 2*A000041(n) for n>0.
Row sums give A058696(n) or A000041(2n).
Cf. A000290, A000712, A001105, A048574, A052837, A304662, A304790.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> x^`if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
          `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n$2, [])):
seq(T(n), n=0..11);

Sum_{k>0} k * T(n,k) = A304662(n).
T(n,A304790(n)) = 1 for n in { A001105 }.
Sum_{k>=0} T(n,k) = A058696(n) = A000041(2n).
Sum_{k>=1} T(n,k) = A000712(n).
Sum_{k>=2} T(n,k) = A048574(n) = A052837(n).

A341236 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^10.

Original entry on oeis.org

1, 20, 210, 1550, 9055, 44624, 192945, 751480, 2686155, 8934560, 27946335, 82884860, 234636435, 637416140, 1669127130, 4228739712, 10398140075, 24882425770, 58080468790, 132508486900, 296005537183, 648445364080, 1394961003490, 2950516502980, 6142674032345, 12599932782780

Offset: 10

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..10000

Column k=10 of A060642.

Cf. A000041, A023009, A048574, A327388, A341221, A341222, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..35);  # Alois P. Heinz, Feb 07 2021

nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

A052837 Number of partitions of 2n whose Ferrers-Young diagram allows more than one different domino tiling.

Original entry on oeis.org

0, 0, 1, 4, 10, 22, 43, 80, 141, 240, 397, 640, 1011, 1568, 2395, 3604, 5360, 7876, 11460, 16510, 23588, 33418, 47006, 65640, 91085, 125596, 172215, 234820, 318579, 430060, 577920, 773130, 1030007, 1366644, 1806445, 2378892, 3121835, 4082796, 5322360, 6916360

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The original name was: A simple grammar.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 804

Essentially the same as A048574.

Cf. A000712, A304789.

spec := [S,{C=Sequence(Z,1 <= card),B=Set(C,1 <= card),S=Prod(B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
a:= n-> (p-> add(p(j)*p(n-j), j=1..n-1))(combinat[numbpart]):
seq(a(n), n=0..40);  # Alois P. Heinz, May 26 2018

a[n_] := If[n <= 1, 0, With[{pp = Array[PartitionsP, n-1]},
   First[ListConvolve[pp, pp]]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 30 2025 *)

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

a(n) = Sum_{k>=2} A304789(n,k). - Alois P. Heinz, May 26 2018

More terms from Franklin T. Adams-Watters, Feb 08 2006

New name from Alois P. Heinz, May 26 2018

A132993 Triangle t(n,m) = P(n-m+1) * P(m+1) read by rows, 0<=m<=n, where P=A000041 are the partition numbers.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 6, 6, 5, 7, 10, 9, 10, 7, 11, 14, 15, 15, 14, 11, 15, 22, 21, 25, 21, 22, 15, 22, 30, 33, 35, 35, 33, 30, 22, 30, 44, 45, 55, 49, 55, 45, 44, 30, 42, 60, 66, 75, 77, 77, 75, 66, 60, 42, 56, 84, 90, 110, 105, 121, 105, 110, 90, 84, 56

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 27 2008

			1;
2, 2;
3, 4, 3;
5, 6, 6, 5;
7, 10, 9, 10, 7;
11, 14, 15, 15, 14, 11;
15, 22, 21, 25, 21, 22, 15;
22, 30, 33, 35, 35, 33, 30, 22;
30, 44, 45, 55, 49, 55, 45, 44, 30;
42, 60, 66, 75, 77, 77, 75, 66, 60, 42;
56, 84, 90, 110, 105, 121, 105, 110, 90, 84, 56;

Cf. A000041, A048574 (row sums).

A132993 := proc(n,m)
        combinat[numbpart](n-m+1)*combinat[numbpart](m+1) ;
end proc:
seq(seq(A132993(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Nov 11 2011

<< DiscreteMath`Combinatorica`; << DiscreteMath`IntegerPartitions`; Clear[t, n, m]; t[n_, m_] = PartitionsP[n - m + 1]*PartitionsP[m + 1]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

cp's OEIS Frontend

A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A023626 Self-convolution of (1, p(1), p(2), ...).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A304789 Number T(n,k) of partitions of 2n whose Ferrers-Young diagram allows exactly k different domino tilings; triangle T(n,k), n>=0, 0<=k<=A304790(n), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A341236 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^10.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A052837 Number of partitions of 2n whose Ferrers-Young diagram allows more than one different domino tiling.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A132993 Triangle t(n,m) = P(n-m+1) * P(m+1) read by rows, 0<=m<=n, where P=A000041 are the partition numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica