A184731
a(n) = Sum_{k=0..n} C(n,k)^(k+1).
Original entry on oeis.org
1, 2, 6, 38, 490, 14152, 969444, 140621476, 46041241698, 36363843928316, 62022250535177416, 236043875222171125276, 2205302277098968939256248, 45728754995013679582534494332, 2070631745797418828103776968679204
Offset: 0
The terms begin:
a(0) = 1;
a(1) = 1 + 1^2 = 2;
a(2) = 1 + 2^2 + 1^3 = 6;
a(3) = 1 + 3^2 + 3^3 + 1^4 = 38;
a(4) = 1 + 4^2 + 6^3 + 4^4 + 1^5 = 490;
a(5) = 1 + 5^2 + 10^3 + 10^4 + 5^5 + 1^6 = 14152.
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Table[Sum[Binomial[n, k]^(k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 29 2014 *)
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{a(n)=sum(k=0, n, binomial(n, k)^(k+1))}
A206151
G.f.: exp( Sum_{n>=1} A206152(n)*x^n/n ), where A206152(n) = Sum_{k=0..n} binomial(n,k)^(n+k).
Original entry on oeis.org
1, 2, 7, 120, 16257, 22426576, 181974299842, 15238138790731690, 8413234043413844801094, 36597622942948070873495055416, 1557743574279376981523155294991683637, 377269728353963189455845962558983304322979834
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 120*x^3 + 16257*x^4 + 22426576*x^5 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 10*x^2/2 + 326*x^3/3 + 64066*x^4/4 + 111968752*x^5/5 +...+ A206152(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(m+k))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))
A219207
Triangle, read by rows, where T(n,k) = binomial(n,k)^(k+1) for n>=0, k=0..n.
Original entry on oeis.org
1, 1, 1, 1, 4, 1, 1, 9, 27, 1, 1, 16, 216, 256, 1, 1, 25, 1000, 10000, 3125, 1, 1, 36, 3375, 160000, 759375, 46656, 1, 1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1, 1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1, 1, 81, 46656
Offset: 0
Triangle of coefficients C(n,k)^(k+1) begins:
1;
1, 1;
1, 4, 1;
1, 9, 27, 1;
1, 16, 216, 256, 1;
1, 25, 1000, 10000, 3125, 1;
1, 36, 3375, 160000, 759375, 46656, 1;
1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1;
1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1; ...
MATRIX INVERSE.
The matrix inverse starts
1;
-1,1;
3,-4,1;
-73,99,-27,1;
18055,-24496,6696,-256,1;
-55694851,75563975,-20656000,790000,-3125,1; - _R. J. Mathar_, Mar 22 2013
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Table[Binomial[n,k]^(k+1),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 15 2016 *)
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{T(n,k)=binomial(n,k)^(k+1)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
A206153
G.f.: exp( Sum_{n>=1} A206154(n)*x^n/n ), where A206154(n) = Sum_{k=0..n} binomial(n,k)^(k+2).
Original entry on oeis.org
1, 2, 7, 48, 693, 26632, 2542514, 533442978, 278979307990, 343728261289376, 904762216681139381, 5771110378770242683658, 88742047516327429085056353, 2912737209806573079629325613400, 224604736339682169442980060945290802
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 + 14921404*x^6/6 +...+ A206154(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(k+2))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))
A206155
G.f.: exp( Sum_{n>=1} A206156(n)*x^n/n ), where A206156(n) = Sum_{k=0..n} binomial(n,k)^(2*k).
Original entry on oeis.org
1, 2, 5, 38, 1425, 283002, 448468978, 2707673843860, 67018498701021670, 14506787732148113566364, 13603174532364904984495776225, 43960529641219941452921634596223366, 1207327102995668834632770987833295579308107, 188859837731175560954429490131760211759694331013582
Offset: 0
G.f.: A(x) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 + 2687407464*x^6/6 +...+ A206156(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(2*k-0))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))
A206157
G.f.: exp( Sum_{n>=1} A206158(n)*x^n/n ), where A206158(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).
Original entry on oeis.org
1, 2, 7, 102, 6261, 2423430, 6686021554, 61335432894584, 2941073857435300366, 1190520035262419577871332, 1696475310227140760623646031573, 9980324833243234634513255755001535870, 565171444566758371735408026461987217216896790
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 + 40086916024*x^6/6 +...+ A206158(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(2*k+1))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))
A228899
Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^(k+1) * y^k ), as read by rows.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 6, 12, 1, 1, 10, 71, 76, 1, 1, 15, 281, 2153, 701, 1, 1, 21, 861, 29166, 129509, 8477, 1, 1, 28, 2212, 244725, 7664343, 12391414, 126126, 1, 1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1, 1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1
Offset: 0
This triangle begins:
1;
1, 1;
1, 3, 1;
1, 6, 12, 1;
1, 10, 71, 76, 1;
1, 15, 281, 2153, 701, 1;
1, 21, 861, 29166, 129509, 8477, 1;
1, 28, 2212, 244725, 7664343, 12391414, 126126, 1;
1, 36, 4998, 1477391, 218030412, 3875325345, 1699148352, 2223278, 1;
1, 45, 10242, 7017577, 3748460115, 448713017405, 3284369541969, 315158247170, 45269999, 1; ...
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+12*y^2+y^3)*x^3 + (1+10*y+71*y^2+76*y^3+y^4)*x^4 + (1+15*y+281*y^2+2153*y^3+701*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x
+ (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^2*y + 3^3*y^2 + y^3)*x^3/3
+ (1+ 4^2*y + 6^3*y^2 + 4^4*y^3 + x^4)*x^4/4
+ (1+ 5^2*y + 10^3*y^2 + 10^4*y^3 + 5^5*y^4 + y^5)*x^5/5
+ (1+ 6^2*y + 15^3*y^2 + 20^4*y^3 + 15^5*y^4 + 6^6*y^5 + y^6)*x^6/6 +...
in which the coefficients form A219207(n,k) = binomial(n, k)^(k+1).
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{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m, j)^(j+1)*y^j))+x*O(x^n)), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
Showing 1-7 of 7 results.
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