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Search: a334549 -id:a334549
Displaying 1-10 of 10 results found. page 1
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A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n. +10
6
1, 1, 7, 31, 175, 991, 5881, 35617, 219871, 1376095, 8710537, 55644337, 358198369, 2320792657, 15120204295, 98984058271, 650725327231, 4293779332927, 28425752310361, 188739799967425, 1256510215733185, 8385127334900305, 56078904057164215, 375796823748323215 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Inverse binomial transform of the Franel numbers (A000172). - Paul D. Hanna, Feb 26 2012
a(n) is the constant term in the expansion of (1 + x + y + 1/x + 1/y + x/y + y/x)^n. - Seiichi Manyama, Oct 26 2019
a(n) is the constant term in the expansion of (-1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019
a(n) is the number of n step closed walks on the hexagonal lattice with loops at each node. A step along a loop leaves the position unchanged. The bijection is as follows: after subtracting 1 from each element in the array, values are -1, 0 or 1 and row and column sums are zero. There are only seven possibilities for each row. An all zero row corresponds with a step along the loop leaving the position unchanged and the others to a unit step in each of the six possible directions. This justifies that this sequence is the binomial transform of A002898. - Andrew Howroyd, May 09 2020
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500 (terms 1..99 from R. H. Hardin)
FORMULA
From Paul D. Hanna, Feb 26 2012: (Start)
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).
Equals the binomial transform of A002898.
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * A000172(k), where A000172(k) = Sum_{j=0..k} binomial(k,j)^3 forms the Franel numbers.
(End)
Recurrence: n^2*a(n) = (2*n-1)^2*a(n-1) + 19*(n-1)^2*a(n-2) + 14*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - Vaclav Kotesovec, Oct 20 2012
G.f.: hypergeom([1/3, 1/3],[1],-27*x*(x+1)^2/((1-7*x)^2*(1+2*x)))/((1+2*x)^(1/3)*(1-7*x)^(2/3)). - Mark van Hoeij, May 07 2013
EXAMPLE
G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...
G.f.: A(x) = 1/(1-x) + 6*x^2*(1+x)/(1-x)^4 + 90*x^4*(1+x)^2/(1-x)^7 + 1680*x^6*(1+x)^3/(1-x)^10 + 34650*x^8*(1+x)^4/(1-x)^13 +...+ A006480(n)*x^(2*n)*(1+x)^n/(1-x)^(3*n+1) +...
MATHEMATICA
Table[SeriesCoefficient[Sum[(3*k)!/k!^3*x^(2*k)*(1+x)^k/(1-x)^(3*k+1), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, (3*m)!/m!^3*x^(2*m)*(1+x)^m/(1-x + x*O(x^n))^(3*m+1)), n)} \\ Paul D. Hanna, Feb 26 2012
(PARI) a(n)={sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3))} \\ Andrew Howroyd, May 09 2020
CROSSREFS
Column k=3 of A328747 and A334549.
Cf. A000172, A002898, A006480, A328725.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
EXTENSIONS
a(0)=1 prepended by Andrew Howroyd, May 09 2020
STATUS
approved
A172645 Number of n X n 0..2 arrays with row sums n and column sums n. +10
3
1, 3, 31, 2371, 1084851, 3680774301, 91358224634433, 17470072106054582211, 26088526624958727703324771, 310419652143758898175543447421953, 29785621316391113552729016416250323294253 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..14
CROSSREFS
Main diagonal of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172635 Number of n X 14 0..2 arrays with row sums 14 and column sums n. +10
2
1, 616227, 15120204295, 2411829493241299, 358060848206529563811, 75490918169569448369461821, 17949532724881770551236183389177, 4867117722741777809293028167592172435 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..14
CROSSREFS
Column k=14 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172636 Number of n X 7 0..2 arrays with row sums 7 and column sums n. +10
2
1, 393, 35617, 7343449, 1509893001, 360255871641, 91358224634433, 24567498558526617, 6887820148172502169, 1998635481349606292593, 596268326872362183077193, 182062424087215419645579529 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..47
CROSSREFS
Column k=7 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172637 Number of n X 10 0..2 arrays with row sums 10 and column sums n. +10
2
1, 8953, 8710537, 30189264889, 101471705778601, 431222728237019041, 1998635481349606292593, 10132345044562368521279737, 54649595712626140857519780409, 310419652143758898175543447421953 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..25
CROSSREFS
Column k=10 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172638 Number of n X 8 0..2 arrays with row sums 8 and column sums n. +10
2
1, 1107, 219871, 115321795, 59794281891, 37015866368181, 24567498558526617, 17470072106054582211, 13039980402350138484115, 10132345044562368521279737, 8132104885895774387439086637, 6705797924085107855686463036869 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..37
CROSSREFS
Column k=8 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172640 Number of n X 12 0..2 arrays with row sums 12 and column sums n. +10
2
1, 73789, 358198369, 8365213132981, 185658731463324901, 5520471267708881730181, 182062424087215419645579529, 6705797924085107855686463036869, 266324891937340036349722115366403541 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..18
CROSSREFS
Column k=12 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172641 Number of n X 9 0..2 arrays with row sums 9 and column sums n. +10
2
1, 3139, 1376095, 1849858771, 2435292751411, 3938448319935781, 6887820148172502169, 13039980402350138484115, 26088526624958727703324771, 54649595712626140857519780409, 118790606468949916540947848175709 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..30
CROSSREFS
Column k=9 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172643 Number of n X 13 0..2 arrays with row sums 13 and column sums n. +10
2
1, 212941, 2320792657, 141444270208645, 8107975032968711941, 641062514242765723819621, 56688799539093817446748651801, 5657362648396875453954444188314837 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..16
CROSSREFS
Column k=13 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
A172644 Number of n X 11 0..2 arrays with row sums 11 and column sums n. +10
2
1, 25653, 55644337, 499616140189, 4307041366439901, 48321459222257750541, 596268326872362183077193, 8132104885895774387439086637, 118790606468949916540947848175709, 1838094093862096792566838713252275953 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
R. H. Hardin, Table of n, a(n) for n=1..20
CROSSREFS
Column k=11 of A334549.
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 06 2010
STATUS
approved
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Last modified August 30 11:14 EDT 2024. Contains 375543 sequences. (Running on oeis4.)