A195836 -id:A195836 - cp's OEIS frontend

A195843 Triangle read by rows which arises from A195833, in the same way as A175003 arises from A195310. Column k starts at row A195313(k).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized tridecagonal numbers A195313, A195833 and A196933 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Row sums give A196933.

Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840, A195841, A195842.

A210954 Triangle read by rows which arises from A210944 in the same way as A175003 arises from A195310. Column k starts at row A195818(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1

Offset: 1

Omar E. Pol, Jun 16 2012

The sum of terms of row n is equal to the leftmost term of row n+1. Also 1 together with the row sums give A210964. This sequence is related to the generalized 14-gonal numbers A195818, A210954 and A210964 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as an irregular triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1;
1;
1,  1;
2,  1;
3,  1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  2, -1;
5,  3, -1;
7,  4, -1;
10, 4, -2;
12, 4, -3;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 5, -4;
14, 7, -4, -1;

Cf. A175003, A195818, A195825, A195836, A195837, A195838, A195839, A195840, A195841, A195842, A195843, A210944, A210964.

A233758 Bisection of A006950 (the even part).

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 28, 43, 70, 105, 161, 236, 350, 501, 722, 1016, 1431, 1981, 2741, 3740, 5096, 6868, 9233, 12306, 16357, 21581, 28394, 37128, 48406, 62777, 81182, 104494, 134131, 171467, 218607, 277691, 351841, 444314, 559727, 703002, 880896, 1100775

Offset: 1

Omar E. Pol, Jan 11 2014

nonn

See Zaletel-Mong paper, page 14, FIG. 11: C2a is this sequence, C2b is A233759, C2c is A015128.

M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el] (2012), 14 (C2a).

Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233759.

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i - Mod[i, 2]]]]];
a[n_] := b[2 n - 2, 2 n - 2];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)

A233759 Bisection of A006950 (the odd part).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 35, 55, 86, 130, 196, 287, 420, 602, 858, 1206, 1687, 2331, 3206, 4368, 5922, 7967, 10670, 14193, 18803, 24766, 32490, 42411, 55159, 71416, 92152, 118434, 151725, 193676, 246491, 312677, 395537, 498852, 627509, 787171, 985043, 1229494

Offset: 1

Omar E. Pol, Jan 11 2014

nonn

See Zaletel-Mong paper, page 14, FIG. 11: C2a is A233758, C2b is this sequence, C2c is A015128.

M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el] (2012), 14 (C2b).

Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233758.

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i - Mod[i, 2]]]]];
a[n_] := b[2 n - 1, 2 n - 1];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)

A233969 Partial sums of A006950.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 17, 24, 34, 47, 63, 84, 112, 147, 190, 245, 315, 401, 506, 636, 797, 993, 1229, 1516, 1866, 2286, 2787, 3389, 4111, 4969, 5985, 7191, 8622, 10309, 12290, 14621, 17362, 20568, 24308, 28676, 33772, 39694, 46562, 54529, 63762, 74432, 86738

Offset: 0

Omar E. Pol, Jan 12 2014

nonn

The first three columns of A211970 are A211971, A000041, A006950, so for k = 0..2, the partial sums of column k of A211970 give: A015128, A000070, this sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A000070, A015128, A195825, A195826, A195836, A210843, A211970, A211971, A233758, A233759.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
    end:
a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 12 2014

Accumulate[CoefficientList[Series[x*QPochhammer[-1/x, x^2]/((1 + x) * QPochhammer[x^2]), {x, 0, 50}], x]] (* Vaclav Kotesovec, Oct 27 2016 *)

a(n) ~ exp(Pi*sqrt(n/2))/(2*Pi*sqrt(n)). - Vaclav Kotesovec, Oct 27 2016

cp's OEIS Frontend

A195843 Triangle read by rows which arises from A195833, in the same way as A175003 arises from A195310. Column k starts at row A195313(k).

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A210954 Triangle read by rows which arises from A210944 in the same way as A175003 arises from A195310. Column k starts at row A195818(k).

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A233758 Bisection of A006950 (the even part).

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A233759 Bisection of A006950 (the odd part).

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A233969 Partial sums of A006950.

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