A197870 -id:A197870 - cp's OEIS frontend

A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

0, 1, 2, 1, 2, 0, 1, 3, 3, 2, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Paul Curtz, Mar 11 2015

Consider the difference table of a sequence with A000004(n)=0's as main diagonal. (Example: A000045(n).) We call this sequence an autosequence of the first kind.
Based on Pascal's triangle, a(n) =
0,                   T1
1, 2,
1, 2, 0,
1, 3, 3, 2,
etc.
transforms every sequence s(n) in an autosequence of the first kind via the multiplication by the triangle
s0,                  T2
s0, s1,
s0, s1, s2,
s0, s1, s2, s3,
etc.
Examples.
1) s(n) = A198631(n)/A006519(n+1), the second fractional Euler numbers (see A209308). This yields 0*1, 1*1+2*1/2=2, 1*1+2*1/2+0*0=2, 1*1+3*1/2++3*0+2*(-1/4)=2, ... .
The autosequence is 0 followed by 2's or 2*(0,1,1,1,1,1,1,1,... = b(n)).
b(n), the basic autosequence of the first kind, is not in the OEIS (see A140575 and A054977).
2) s(n) = A164555(n)/A027642(n), the second Bernoulli numbers, yields 0,2,2,3,4,5,6,7,... = A254667(n).
Row sums of T1: A062510(n) = 3*A001045(n).
Antidiagonal sums of T1: A111573(n).
With 0's instead of the spaces, every column, i.e.,
0, 1, 1, 1, 1,  1,  1,  1,  1,  1,   1, ...
0, 2, 2, 3, 4,  5,  6,  7,  8,  9,  10, ... = A001477(n) with 0 instead of 1 = A254667(n)
0, 0, 0, 3, 6, 10, 15, 21, 28, 36,  45, ... = A161680(n) with 0 instead of 1
0, 0, 0, 2, 4, 10, 20, 35, 56, 84, 120, ...
etc., is an autosequence of the first kind.
With T(0,0) = 1, it is (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 24 2015

			Triangle starts:
0;
1, 2;
1, 2, 0;
1, 3, 3, 2;
1, 4, 6, 4, 0;
1, 5, 10, 10, 5, 2;
1, 6, 15, 20, 15, 6, 0;
...

Cf. A001045, A001477, A010673, A011973, A054977, A062510, A007318, A197870, A006519, A198631, A140575, A209308, A164555, A027642, A111573, A161680, A254667.

a[n_, k_] := If[k == n, 2*Mod[n, 2], Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 23 2015 *)

a(n) = Pascal's triangle A007318(n) with main diagonal A010673(n) (= period 2: repeat 0, 2) instead of 1's=A000012(n).
a(n) = reversal abs(A140575(n)).
a(n) = A007318(n) + A197870(n+1).
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k>n or if k<0 . - Philippe Deléham, May 24 2015
G.f.: (-1-2*x*y+x^2*y+x^2*y^2)/((x*y+1)*(x*y+x-1)) - 1. - R. J. Mathar, Aug 12 2015

A006330 Number of corners, or planar partitions of n with only one row and one column.

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 38, 63, 106, 170, 272, 422, 653, 986, 1482, 2191, 3218, 4666, 6726, 9592, 13602, 19122, 26733, 37102, 51232, 70292, 95989, 130356, 176246, 237120, 317724, 423840, 563266, 745562, 983384, 1292333, 1692790, 2209886, 2876132

Offset: 0

N. J. A. Sloane

nonn

The first four terms a(0), a(1), a(2), a(3) agree with sequence A000219 for unrestricted planar partitions, since the restriction does not rule anything out. For a(4) there is just one planar partition which doesn't satisfy the restriction, four 1's arranged in a square. So A000219 has fifth term 13 and here we have 12.
a(n) + A001523(n) = A000712(n). - Michael Somos, Jul 22 2003
Number of unimodal compositions of n+1 where the maximal part appears once, see example. [Joerg Arndt, Jun 11 2013]

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(4)=12 unimodal compositions of 4+1=5 where the maximal part appears once:
01:  [ 1 1 1 2 ]
02:  [ 1 1 2 1 ]
03:  [ 1 1 3 ]
04:  [ 1 2 1 1 ]
05:  [ 1 3 1 ]
06:  [ 1 4 ]
07:  [ 2 1 1 1 ]
08:  [ 2 3 ]
09:  [ 3 1 1 ]
10:  [ 3 2 ]
11:  [ 4 1 ]
12:  [ 5 ]
(End)
G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 21*x^5 + 38*x^6 + 63*x^7 + 106*x^8 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see page 77.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See (5.6).
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)
G. Kreweras, Letter to N. J. A. Sloane

Cf. A000219, A000712, A197870.
Column k=1 of A247255.
Row sums of A259100.

a[0] = 1; a[n_] := SeriesCoefficient[ Sum[x^k/Product[1 - x^i, {i, 1, k}]^2, {k, 1, n}] + 1, {x, 0, n}]; Array[a, 39, 0] (* Jean-François Alcover, Mar 13 2014 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / prod(i=1, k, 1 - x^i, 1 + x*O(x^n))^2, 1), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) - 1)\2, (-1)^k * x^((k + k^2)/2)) / eta(x + x*O(x^n))^2, n))};

G.f.: 1+Sum_{k>0} x^k/(Product_{i=1..k} (1-x^i))^2.
G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1 - x^k)^2. - Michael Somos, Jul 28 2003
Convolution product of A197870 and A000712. - Michael Somos, Feb 22 2015
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015

Edited and extended by Moshe Shmuel Newman, Jun 10 2003

A208061 G.f. 1/sum(k>=0, (-1)^k * x^(k*(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 0, -1, -2, -1, 1, 4, 5, 2, -5, -12, -13, -3, 17, 34, 32, -1, -54, -93, -72, 28, 169, 248, 152, -147, -510, -646, -282, 582, 1484, 1627, 375, -2045, -4195, -3927, 110, 6716, 11544, 9002, -3458, -20996, -30921, -19123, 17974, 63154, 80435, 35553, -71525, -183969

Offset: 0

Franklin T. Adams-Watters, Feb 23 2012

sign

			G.f. = 1 + x + x^2 - x^4 - 2*x^5 - x^6 + x^7 + 4*x^8 + 5*x^9 + 2*x^10 - 5*x^11 + ...

Cf. A006950, A023361, A106507, A197870.

al(n)=Vec(1/(sum(k=0,sqrtint(2*n),(-1)^k*x^(k*(k+1)\2))+x*O(x^n)))

G.f.: 1 / (1 - x*(1 - x^2*(1 - x^3*(1 - x^4*(1 - ...))))). - Michael Somos, Mar 03 2014

Convolution inverse of A197870. - Michael Somos, Mar 03 2014

A277078 Triangular array similar to A255935 but with 0's and 2's swapped in the trailing diagonal. The columns alternate in signs.

Original entry on oeis.org

2, 1, 0, 1, -2, 2, 1, -3, 3, 0, 1, -4, 6, -4, 2, 1, -5, 10, -10, 5, 0, 1, -6, 15, -20, 15, -6, 2, 1, -7, 21, -35, 35, -21, 7, 0, 1, -8, 28, -56, 70, -56, 28, -8, 2, 1, -9, 36, -84, 126, -126, 84, -36, 9, 0, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 2

Offset: 0

Paul Curtz, Oct 23 2016

a(n)=
2,
1,  0,
1, -2, 2,
1, -3, 3,  0,
1, -4, 6, -4, 2,
etc.
transforms every sequence s(n) in an autosequence of the second kind via the multiplication by the triangle
s0,           T2
s0, s1,
s0, s1, s2,
s0, s1, s2, s3,
etc.
which is the reluctant form of s(n).
Example.
s(n) = A131577(n) = 0, 1, 2, 4, ... .
The multiplication gives 0, 0, 2, 3, 8, 15, 32, 63, ... = 0 followed by A166920.
a(n) comes from alternate sum and difference of s(n) and t(n), its inverse binomial transform. In the example (t(n) = periodic 2: repeat 0, 1) the first terms are: 0+0, 1-1, 2+0, 4-1, 8+0, 16-1, 32+0, 64-1, ... .

Cf. A000035, A007318, A054977, A131577, A166920, A197870, A255935.

a[n_, k_] := If[k == n, 2 - 2*Mod[n, 2], (-1)^k*Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 16 2016 *)

a(n) = A007318(n) - A197870(n+1).

cp's OEIS Frontend

A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

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A006330 Number of corners, or planar partitions of n with only one row and one column.

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A208061 G.f. 1/sum(k>=0, (-1)^k * x^(k*(k+1)/2)).

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A277078 Triangular array similar to A255935 but with 0's and 2's swapped in the trailing diagonal. The columns alternate in signs.

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