A122400
Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.
Original entry on oeis.org
1, 1, 4, 31, 338, 4769, 82467, 1687989, 39905269, 1069863695, 32071995198, 1062991989013, 38596477083550, 1523554760656205, 64961391010251904, 2975343608212835855, 145687881987604377815, 7594435556630244257213
Offset: 0
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A122399 := proc(n) option remember ; add( combinat[stirling2](n,k)*k^n*k!,k=0..n) ; end: A122400 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122399(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122400(n)) ; od ; # R. J. Mathar, May 18 2007
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max = 17; CoefficientList[ Series[ 1 + Sum[ ((1 + x)^n - 1)^n, {n, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 26 2013, after Vladeta Jovovic *)
A187826
G.f. satisfies: A(x) = Sum_{n>=0} ((1 + x*A(x))^n - 1)^n / (1 + x*A(x))^(n^2).
Original entry on oeis.org
1, 1, 4, 26, 219, 2227, 26438, 359904, 5555201, 96383191, 1864908541, 39929905561, 938897407239, 24069888638463, 668309231078015, 19977542570492051, 639571311256259372, 21828488143257352752, 791044181963746918758, 30331001954496565907536
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 219*x^4 + 2227*x^5 + 26438*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x*A(x)/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)^2/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^3/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^4/(1+x*A(x))^16 +...
(2) A(x) = 1/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^2/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^3/(1+x*A(x))^16 +...
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{a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
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{a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
A220352
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n^2).
Original entry on oeis.org
1, 1, 3, 16, 118, 1116, 12869, 175096, 2745726, 48756438, 967026762, 21188546616, 508286084222, 13249410224210, 372908807794347, 11270832179901016, 364083312029454453, 12518063823862065816, 456432182550333723335, 17591590487681007523476
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1116*x^5 + 12869*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x/(1+x) + (2*x + x^2)^2/(1+x)^4 + (3*x + 3*x^2 + x^3)^3/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^4/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^5/(1+x)^25 +...
(2) A(x) = 1/(1+x) + (2*x + x^2)/(1+x)^4 + (3*x + 3*x^2 + x^3)^2/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^3/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^4/(1+x)^25 +...
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{a(n)=local(q=1+x+x*O(x^n),A=1);A=sum(k=0,n,q^(-k^2)*(q^k-1)^k);polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
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{a(n)=local(q=1+x+x*O(x^n),A=1);A=sum(k=1,n+1,q^(-k^2)*(q^k-1)^(k-1));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
A187827
G.f. satisfies: A(x) = Sum_{n>=0} (1 - (1 - x*A(x))^n)^n.
Original entry on oeis.org
1, 1, 5, 36, 325, 3468, 42519, 590268, 9201740, 160150252, 3095440553, 66068011710, 1547572760559, 39529002357409, 1094096683131616, 32622859912512090, 1042350065213470532, 35521574976088978133, 1285782300453328211074, 49256935742079848796102
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 36*x^3 + 325*x^4 + 3468*x^5 + 42519*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x*A(x) + (1 - (1-x*A(x))^2)^2 + (1 - (1-x*A(x))^3)^3 + (1 - (1-x*A(x))^4)^4 + (1 - (1-x*A(x))^5)^5 +...
(2) A(x) = (1-x*A(x)) + (1-x*A(x))^2*(1 - (1-x*A(x))^2) + (1-x*A(x))^3*(1 - (1-x*A(x))^3)^2 + (1-x*A(x))^4*(1 - (1-x*A(x))^4)^3 + (1-x)^5*(1 - (1-x*A(x))^5)^4 +...
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{a(n)=local(q, A=1); for(i=1,n,q=1/(1-x*A+x*O(x^n));A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
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{a(n)=local(q, A=1); for(i=1,n,q=1/(1-x*A+x*O(x^n));A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
Showing 1-4 of 4 results.
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