Search: seq:1,1,0,1,1,0,1,3,2,0
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A322114
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Regular triangle read by rows where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.
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+30
17
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1, 1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 0, 3, 6, 3, 0, 0, 2, 11, 14, 6, 0, 0, 1, 13, 35, 33, 11, 0, 0, 0, 10, 61, 112, 81, 23, 0, 0, 0, 5, 75, 262, 347, 204, 47, 0, 0, 0, 2, 68, 463, 1059, 1085, 526, 106, 0, 0, 0, 1, 49, 625, 2458, 4091, 3348, 1376, 235
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OFFSET
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0,9
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LINKS
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EXAMPLE
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Triangle begins:
1
1 1
0 1 1
0 1 3 2
0 0 3 6 3
0 0 2 11 14 6
0 0 1 13 35 33 11
Non-isomorphic representatives of the graphs counted in row 4:
{{2}{3}{12}{13}} {{4}{12}{23}{34}} {{13}{24}{35}{45}}
{{2}{3}{13}{23}} {{4}{13}{23}{34}} {{14}{25}{35}{45}}
{{3}{12}{13}{23}} {{4}{13}{24}{34}} {{15}{25}{35}{45}}
{{4}{14}{24}{34}}
{{12}{13}{24}{34}}
{{14}{23}{24}{34}}
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PROG
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(PARI)
InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i), 1))}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n, 1..n]))} \\ Andrew Howroyd, Oct 22 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A350448
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Triangle read by rows: T(n,k) is the number of acyclic graphs on n unlabeled nodes whose longest directed path has k arcs.
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+30
2
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1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 8, 14, 8, 0, 1, 20, 89, 128, 64, 0, 1, 55, 634, 1934, 2336, 1024, 0, 1, 163, 5668, 36428, 83648, 84992, 32768, 0, 1, 556, 67926, 959718, 3919584, 7097088, 6144000, 2097152, 0, 1, 2222, 1137641, 37205922, 268989920, 793138688, 1175224320, 880803840, 268435456, 0
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OFFSET
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0,8
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LINKS
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EXAMPLE
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Triangle begins:
1;
1, 0;
1, 1, 0;
1, 3, 2, 0;
1, 8, 14, 8, 0;
1, 20, 89, 128, 64, 0;
1, 55, 634, 1934, 2336, 1024, 0;
1, 163, 5668, 36428, 83648, 84992, 32768, 0;
...
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PROG
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(PARI) \\ See PARI link in A122078 for program code.
{ my(T=AcyclicDigraphsByLongestPath(8)); for(n=1, #T, print(T[n])) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A355423
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Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{j=1..k} (exp(j*x) - 1)).
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+30
2
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1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 6, 14, 5, 0, 1, 10, 50, 81, 15, 0, 1, 15, 130, 504, 551, 52, 0, 1, 21, 280, 2000, 5870, 4266, 203, 0, 1, 28, 532, 6075, 35054, 76872, 36803, 877, 0, 1, 36, 924, 15435, 148429, 684000, 1111646, 348543, 4140, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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T(0,k) = 1 and T(n,k) = Sum_{i=1..n} (Sum_{j=1..k} j^i) * binomial(n-1,i-1) * T(n-i,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 3, 6, 10, 15, ...
0, 2, 14, 50, 130, 280, ...
0, 5, 81, 504, 2000, 6075, ...
0, 15, 551, 5870, 35054, 148429, ...
0, 52, 4266, 76872, 684000, 4004100, ...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A054654
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Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.
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+20
12
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1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 11, -6, 0, 1, -10, 35, -50, 24, 0, 1, -15, 85, -225, 274, -120, 0, 1, -21, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0
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OFFSET
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0,8
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COMMENTS
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Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.
Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x. E.g., 3!*C(x,3) = x^3-3*x^2+2*x.
The matrix product of binomial coefficients with the Stirling numbers of the first kind results in the Stirling numbers of the first kind again, but the triangle is shifted by (1,1).
Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] where DELTA is the operator defined in A084938; mirror image of the Stirling-1 triangle A048994. - Philippe Deléham, Dec 30 2006
The sum of the entries on each row of the triangle, starting on the 3rd row, equals 0. E.g., 1+(-3)+2+0 = 0.
The entries on the triangle can be computed as follows. T(n,r) = T(n-1,r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) = 1 if n==1. (End)
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LINKS
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FORMULA
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n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
(In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.
T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021
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EXAMPLE
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Matrix begins:
1, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, -1, 2, -6, 24, -120, 720, -5040, ...
0, 0, 1, -3, 11, -50, 274, -1764, 13068, ...
0, 0, 0, 1, -6, 35, -225, 1624, -13132, ...
0, 0, 0, 0, 1, -10, 85, -735, 6769, ...
0, 0, 0, 0, 0, 1, -15, 175, -1960, ...
0, 0, 0, 0, 0, 0, 1, -21, 322, ...
0, 0, 0, 0, 0, 0, 0, 1, -28, ...
0, 0, 0, 0, 0, 0, 0, 0, 1, ...
...
Triangle begins:
1;
1, 0;
1, -1, 0;
1, -3, 2, 0;
1, -6, 11, -6, 0;
1, -10, 35, -50, 24, 0;
1, -15, 85, -225, 274, -120, 0;
1, -21, 175, -735, 1624, -1764, 720, 0;
...
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MAPLE
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a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x, n)), x, n-k), k=0..n) end: # Peter Luschny, Nov 28 2010
seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021
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MATHEMATICA
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row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Feb 16 2012, after Maple *)
Table[StirlingS1[n, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Jun 17 2023 *)
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PROG
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(PARI) T(n, k)=polcoeff(n!*binomial(x, n), n-k)
(Haskell)
a054654 n k = a054654_tabl !! n !! k
a054654_row n = a054654_tabl !! n
a054654_tabl = map reverse a048994_tabl
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CROSSREFS
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Essentially Stirling numbers of first kind, multiplied by factorials - see A008276.
The Stirling2 counterpart is A106800.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A322324
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Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Product_{p|n, p prime} (1 - p^k).
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+20
1
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1, 1, 0, 1, -1, 0, 1, -3, -2, 0, 1, -7, -8, -1, 0, 1, -15, -26, -3, -4, 0, 1, -31, -80, -7, -24, 2, 0, 1, -63, -242, -15, -124, 24, -6, 0, 1, -127, -728, -31, -624, 182, -48, -1, 0, 1, -255, -2186, -63, -3124, 1200, -342, -3, -2, 0, 1, -511, -6560, -127, -15624, 7502, -2400, -7, -8, 4, 0
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OFFSET
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1,8
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LINKS
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FORMULA
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G.f. of column k: Sum_{j>=1} mu(j)*j^k*x^j/(1 - x^j).
Dirichlet g.f. of column k: zeta(s)/zeta(s-k).
A(n,k) = Sum_{d|n} mu(d)*d^k.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
0, -1, -3, -7, -15, -31, ...
0, -2, -8, -26, -80, -242, ...
0, -1, -3, -7, -15, -31, ...
0, -4, -24, -124, -624, -3124, ...
0, 2, 24, 182, 1200, 7502, ...
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MATHEMATICA
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Table[Function[k, Product[1 - Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j] j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[d] d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
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PROG
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(PARI) T(n, k) = sumdiv(n, d, moebius(d)*d^k);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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