A225345 -id:A225345 - cp's OEIS frontend

A225338 Number of n X 2 -1,1 arrays such that the sum over i=1..n, j=1..2 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 2-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

1, 1, 1, 1, 3, 7, 15, 33, 77, 181, 443, 1113, 2837, 7283, 18909, 49635, 131427, 350419, 940417, 2538857, 6890577, 18790165, 51462893, 141509487, 390530601, 1081369087, 3003537529, 8366306613, 23366125605, 65420219243, 183585473369, 516298786843, 1454928750641

Offset: 0

R. H. Hardin, May 05 2013

nonn

			All solutions for n=4
.-1..1...-1.-1....1..1
.-1..1....1..1...-1.-1
.-1..1....1..1...-1.-1
.-1..1...-1.-1....1..1

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 1..196 from R. H. Hardin)

Column 2 of A225345.

Cf. A002426.

b:= proc(x, y, t) option remember; `if`(x=0, 1, add(`if`(abs(j)
       b(n, 0$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Mar 25 2020

a(0)=1 prepended by Alois P. Heinz, Mar 25 2020

A225339 Number of nX3 -1,1 arrays such that the sum over i=1..n,j=1..3 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 3-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 0, 0, 6, 0, 0, 0, 410, 0, 0, 0, 47990, 0, 0, 0, 7030132, 0, 0, 0, 1163743294, 0, 0, 0, 208310242542, 0, 0, 0, 39371238885750, 0, 0, 0, 7745062224896578, 0, 0, 0, 1571069270148096124, 0, 0, 0, 326518765865030556374, 0, 0, 0

Offset: 1

R. H. Hardin May 05 2013

nonn

Column 3 of A225345

			All solutions for n=4
..1..1..1...-1.-1.-1...-1..1..1...-1..1..1...-1.-1..1...-1.-1..1
.-1.-1.-1....1..1..1...-1.-1..1...-1.-1.-1....1..1..1...-1..1..1
.-1.-1.-1....1..1..1...-1.-1..1....1..1..1...-1.-1.-1...-1..1..1
..1..1..1...-1.-1.-1...-1..1..1...-1.-1..1...-1..1..1...-1.-1..1

R. H. Hardin, Table of n, a(n) for n = 1..159

A225340 Number of nX4 -1,1 arrays such that the sum over i=1..n,j=1..4 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 4-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

1, 1, 3, 9, 31, 107, 395, 1525, 6095, 24893, 103583, 437763, 1874497, 8116459, 35485099, 156453779, 694956997, 3107415563, 13976787615, 63201012187, 287163163723, 1310481411725, 6004334305049, 27611237977705, 127399143883589

Offset: 1

R. H. Hardin May 05 2013

nonn

Column 4 of A225345

			Some solutions for n=4
.-1..1..1..1....1..1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1..1..1..1
.-1.-1.-1.-1...-1.-1.-1.-1....1..1..1..1...-1.-1..1..1...-1.-1.-1..1
.-1..1..1..1...-1.-1.-1.-1....1..1..1..1...-1.-1..1..1...-1.-1.-1..1
.-1.-1..1..1....1..1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..135

A225341 Number of nX5 -1,1 arrays such that the sum over i=1..n,j=1..5 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 5-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 0, 0, 12, 0, 0, 0, 4508, 0, 0, 0, 2680534, 0, 0, 0, 1985837462, 0, 0, 0, 1663045511892, 0, 0, 0, 1506336601246000, 0, 0, 0, 1440828827246863370, 0, 0, 0, 1434551060282150636452, 0, 0, 0, 1472882578482323918604276, 0, 0, 0

Offset: 1

R. H. Hardin May 05 2013

nonn

Column 5 of A225345

			Some solutions for n=4
.-1.-1.-1.-1..1...-1.-1.-1.-1.-1....1..1..1..1..1...-1.-1.-1..1..1
.-1..1..1..1..1....1..1..1..1..1...-1.-1.-1.-1.-1...-1..1..1..1..1
.-1..1..1..1..1....1..1..1..1..1...-1.-1.-1.-1.-1...-1.-1.-1.-1..1
.-1.-1.-1.-1..1...-1.-1.-1.-1.-1....1..1..1..1..1...-1.-1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..123

A225342 Number of nX6 -1,1 arrays such that the sum over i=1..n,j=1..6 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 6-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

1, 1, 3, 17, 81, 397, 2073, 11291, 63121, 360909, 2102597, 12439855, 74561687, 451920441, 2765717847, 17069622035, 106139191511, 664358517275, 4183117802521, 26479526009223, 168426949915891, 1076002461947039

Offset: 1

R. H. Hardin May 05 2013

nonn

Column 6 of A225345

			Some solutions for n=4
.-1.-1.-1..1..1..1...-1.-1.-1..1..1..1...-1.-1.-1.-1..1..1...-1.-1.-1.-1.-1..1
.-1.-1.-1.-1..1..1...-1.-1..1..1..1..1...-1..1..1..1..1..1...-1..1..1..1..1..1
.-1..1..1..1..1..1...-1.-1.-1.-1.-1..1...-1.-1.-1.-1..1..1...-1..1..1..1..1..1
.-1.-1.-1.-1..1..1...-1.-1..1..1..1..1...-1.-1.-1..1..1..1...-1.-1.-1.-1.-1..1

R. H. Hardin, Table of n, a(n) for n = 1..112

A225343 Number of nX7 -1,1 arrays such that the sum over i=1..n,j=1..7 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 7-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 0, 0, 22, 0, 0, 0, 25056, 0, 0, 0, 47084448, 0, 0, 0, 110202390974, 0, 0, 0, 291612344834026, 0, 0, 0, 834664320724959518, 0, 0, 0, 2522954614267660110474, 0, 0, 0, 7938386813443913197221460, 0, 0, 0, 25757964315025150183357153902, 0, 0, 0

Offset: 1

R. H. Hardin May 05 2013

nonn

Column 7 of A225345

			Some solutions for n=4
.-1.-1.-1.-1..1..1..1...-1.-1..1..1..1..1..1...-1.-1.-1.-1.-1..1..1
.-1.-1.-1..1..1..1..1...-1.-1.-1.-1.-1..1..1...-1..1..1..1..1..1..1
.-1.-1.-1..1..1..1..1...-1.-1.-1.-1.-1..1..1...-1.-1.-1.-1..1..1..1
.-1.-1.-1.-1..1..1..1...-1.-1..1..1..1..1..1...-1.-1.-1.-1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..105

A225344 Number of nX8 -1,1 arrays such that the sum over i=1..n,j=1..8 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 8-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

1, 1, 5, 27, 171, 1081, 7261, 50659, 364051, 2676331, 20044255, 152452967, 1174757541, 9154016699, 72023900129, 571498084391, 4568690484153, 36765944370729, 297627015600415, 2422216988266317, 19808260648181603

Offset: 1

R. H. Hardin May 05 2013

nonn

Column 8 of A225345

			Some solutions for n=4
.-1.-1.-1.-1..1..1..1..1...-1.-1.-1.-1..1..1..1..1...-1.-1.-1.-1.-1.-1..1..1
.-1.-1..1..1..1..1..1..1...-1.-1.-1.-1.-1.-1..1..1...-1.-1..1..1..1..1..1..1
.-1.-1.-1.-1.-1.-1.-1.-1....1..1..1..1..1..1..1..1...-1.-1..1..1..1..1..1..1
.-1.-1..1..1..1..1..1..1...-1.-1.-1.-1.-1.-1..1..1...-1.-1.-1.-1.-1.-1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..97

A225346 Number of 6Xn -1,1 arrays such that the sum over i=1..6,j=1..n of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute n-across galley oarsmen left-right at 6 fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 15, 0, 107, 0, 397, 0, 1081, 0, 2399, 0, 4675, 0, 8273, 0, 13641, 0, 21267, 0, 31731, 0, 45637, 0, 63697, 0, 86635, 0, 115291, 0, 150513, 0, 193269, 0, 244527, 0, 305383, 0, 376929, 0, 460389, 0, 556975, 0, 668043, 0, 794921, 0, 939097, 0, 1102023, 0, 1285311

Offset: 1

R. H. Hardin May 05 2013

nonn

Row 6 of A225345

			Some solutions for n=4
.-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1.-1..1..1...-1..1..1..1
.-1.-1.-1..1...-1..1..1..1....1..1..1..1...-1.-1..1..1...-1.-1.-1.-1
.-1.-1.-1..1....1..1..1..1...-1.-1.-1..1....1..1..1..1...-1.-1.-1..1
..1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1...-1.-1.-1.-1....1..1..1..1
.-1.-1.-1.-1....1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1....1..1..1..1
.-1..1..1..1...-1.-1.-1.-1....1..1..1..1....1..1..1..1...-1.-1.-1.-1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = a(n-2) +2*a(n-4) -a(n-6) -2*a(n-8) +a(n-12) -a(n-14) +2*a(n-18) +a(n-20) -2*a(n-22) -a(n-24) +a(n-26)

A225347 Number of 7Xn -1,1 arrays such that the sum over i=1..7,j=1..n of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute n-across galley oarsmen left-right at 7 fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

0, 33, 0, 395, 0, 2073, 0, 7261, 0, 19709, 0, 45385, 0, 92673, 0, 173189, 0, 301799, 0, 497661, 0, 783969, 0, 1189311, 0, 1747059, 0, 2497109, 0, 3484867, 0, 4763439, 0, 6392095, 0, 8439027, 0, 10979255, 0, 14097919, 0, 17887629, 0, 22452391, 0

Offset: 1

R. H. Hardin May 05 2013

nonn

Row 7 of A225345

			Some solutions for n=4
.-1.-1..1..1...-1.-1.-1..1...-1.-1..1..1...-1.-1.-1.-1...-1.-1..1..1
.-1.-1..1..1....1..1..1..1...-1.-1..1..1...-1..1..1..1...-1.-1.-1.-1
.-1.-1.-1..1...-1.-1..1..1...-1.-1..1..1...-1..1..1..1...-1..1..1..1
.-1..1..1..1...-1.-1..1..1....1..1..1..1....1..1..1..1....1..1..1..1
.-1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1...-1.-1.-1.-1...-1.-1.-1..1
.-1.-1.-1..1...-1.-1.-1.-1...-1.-1.-1.-1...-1..1..1..1....1..1..1..1
.-1.-1..1..1....1..1..1..1....1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = a(n-2) +a(n-4) -a(n-10) -2*a(n-14) +a(n-18) +a(n-20) +a(n-22) +a(n-24) -2*a(n-28) -a(n-32) +a(n-38) +a(n-40) -a(n-42)

A225348 Number of 8Xn -1,1 arrays such that the sum over i=1..8,j=1..n of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute n-across galley oarsmen left-right at 8 fore-aft positions so that there are no turning moments on the ship).

Original entry on oeis.org

8, 77, 410, 1525, 4508, 11291, 25056, 50659, 95130, 168289, 283338, 457627, 713374, 1078615, 1587974, 2283803, 3217102, 4448769, 6050622, 8106883, 10715232, 13988475, 18055740, 23064277, 29180742, 36593245, 45512628, 56174753, 68841880

Offset: 1

R. H. Hardin May 05 2013

nonn

Row 8 of A225345

			Some solutions for n=4
.-1..1..1..1...-1..1..1..1...-1..1..1..1...-1..1..1..1...-1.-1..1..1
.-1..1..1..1...-1..1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1.-1.-1..1
.-1.-1..1..1...-1.-1..1..1...-1..1..1..1...-1..1..1..1...-1..1..1..1
.-1.-1.-1.-1...-1.-1.-1.-1...-1.-1.-1..1...-1.-1.-1.-1...-1.-1.-1..1
.-1.-1.-1.-1...-1.-1.-1..1...-1.-1..1..1...-1.-1.-1..1....1..1..1..1
.-1..1..1..1...-1.-1.-1..1...-1..1..1..1...-1.-1..1..1...-1.-1..1..1
.-1.-1.-1..1...-1.-1..1..1....1..1..1..1...-1.-1.-1..1...-1.-1.-1..1
..1..1..1..1....1..1..1..1...-1.-1.-1.-1....1..1..1..1...-1.-1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = a(n-1) +a(n-2) -a(n-5) -a(n-7) -a(n-8) +a(n-10) +a(n-11) +2*a(n-12) -2*a(n-16) -a(n-17) -a(n-18) +a(n-20) +a(n-21) +a(n-23) -a(n-26) -a(n-27) +a(n-28)

cp's OEIS Frontend

A225338 Number of n X 2 -1,1 arrays such that the sum over i=1..n, j=1..2 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 2-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A225339 Number of nX3 -1,1 arrays such that the sum over i=1..n,j=1..3 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 3-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A225340 Number of nX4 -1,1 arrays such that the sum over i=1..n,j=1..4 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 4-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A225341 Number of nX5 -1,1 arrays such that the sum over i=1..n,j=1..5 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 5-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A225342 Number of nX6 -1,1 arrays such that the sum over i=1..n,j=1..6 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 6-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A225343 Number of nX7 -1,1 arrays such that the sum over i=1..n,j=1..7 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 7-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A225344 Number of nX8 -1,1 arrays such that the sum over i=1..n,j=1..8 of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute 8-across galley oarsmen left-right at n fore-aft positions so that there are no turning moments on the ship).

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A225346 Number of 6Xn -1,1 arrays such that the sum over i=1..6,j=1..n of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute n-across galley oarsmen left-right at 6 fore-aft positions so that there are no turning moments on the ship).

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Formula

A225347 Number of 7Xn -1,1 arrays such that the sum over i=1..7,j=1..n of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute n-across galley oarsmen left-right at 7 fore-aft positions so that there are no turning moments on the ship).

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A225348 Number of 8Xn -1,1 arrays such that the sum over i=1..8,j=1..n of i*x(i,j) is zero, the sum of x(i,j) is zero, and rows are nondecreasing (number of ways to distribute n-across galley oarsmen left-right at 8 fore-aft positions so that there are no turning moments on the ship).

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