A360885
G.f. satisfies A(x) = 1 + x * A(x * (1 + x^2)).
Original entry on oeis.org
1, 1, 1, 1, 2, 4, 7, 16, 39, 93, 246, 671, 1884, 5578, 16887, 52854, 170649, 563703, 1914366, 6649798, 23610987, 85689987, 317054427, 1196183592, 4595744763, 17965311672, 71426213637, 288535755417, 1183807706841, 4929801601890, 20825803784129, 89210585925338
Offset: 0
-
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\3, binomial(i-1-2*j, j)*v[i-2*j])); v;
A360894
G.f. satisfies A(x) = 1 + x * A(x * (1 - x)).
Original entry on oeis.org
1, 1, 1, 0, -2, -1, 7, 0, -44, 69, 276, -1471, 675, 20407, -90560, -20552, 2141700, -10558223, -675239, 329376824, -2106253225, 2364924062, 67114942438, -621638176430, 1926931098484, 14768396756732, -236623058229675, 1371752460097440, 1098671590491324
Offset: 0
-
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\2, (-1)^j*binomial(i-1-j, j)*v[i-j])); v;
A127783
G.f. satisfies: A(x) = 1 + x*A(x+x^2)^2.
Original entry on oeis.org
1, 1, 2, 7, 28, 133, 700, 4039, 25160, 167637, 1186482, 8872752, 69810994, 575912978, 4967058182, 44675926159, 418157494016, 4065044047125, 40973402647058, 427535487044903, 4611642948647118, 51354908876927025
Offset: 0
-
{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x+x^2));polcoeff(A,n)}
A360886
G.f. satisfies A(x) = 1 + x * A(x * (1 + x^3)).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 4, 7, 11, 22, 49, 104, 212, 471, 1112, 2584, 6000, 14574, 36488, 91148, 230011, 596893, 1574433, 4171388, 11193376, 30594229, 84527225, 235243027, 662702701, 1891111335, 5443353369, 15797764276, 46336647188, 137245713050, 409670144432
Offset: 0
-
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\4, binomial(i-1-3*j, j)*v[i-3*j])); v;
A127784
G.f. satisfies: A(x) = 1 + (x+x^2)*A(x+x^2)^2.
Original entry on oeis.org
1, 1, 3, 11, 51, 266, 1540, 9681, 65291, 468401, 3551693, 28327029, 236731183, 2066583601, 18796448581, 177735656083, 1743920667437, 17725856560839, 186366309301259, 2024042644283702, 22679125592930412, 261873356070694571
Offset: 0
-
{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+subst(x*A^2,x,x+x^2));polcoeff(A,n)}
A130521
Triangle, read by rows, where T(n,k) = T(n,k-1) + T(n-1,k-2) for n>=k>=2, with T(n+1,1) = T(n+1,0) = T(n,n) and T(0,0) = 1 for n>=0.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 6, 8, 11, 11, 11, 15, 19, 25, 33, 33, 33, 44, 55, 70, 89, 114, 114, 114, 147, 180, 224, 279, 349, 438, 438, 438, 552, 666, 813, 993, 1217, 1496, 1845, 1845, 1845, 2283, 2721, 3273, 3939, 4752, 5745, 6962, 8458, 8458, 8458, 10303
Offset: 0
T(5,3) = T(5,2) + T(4,1) = 15 + 4 = 19;
T(6,4) = T(6,3) + T(5,2) = 55 + 15 = 70;
T(7,0) = T(6,6) = 89 + 25 = 114.
Triangle begins:
1;
1, 1;
1, 1, 2;
2, 2, 3, 4;
4, 4, 6, 8, 11;
11, 11, 15, 19, 25, 33;
33, 33, 44, 55, 70, 89, 114;
114, 114, 147, 180, 224, 279, 349, 438;
438, 438, 552, 666, 813, 993, 1217, 1496, 1845;
1845, 1845, 2283, 2721, 3273, 3939, 4752, 5745, 6962, 8458; ...
A130522
Diagonal immediately below the main diagonal of triangle A130521.
Original entry on oeis.org
1, 1, 3, 8, 25, 89, 349, 1496, 6962, 34861, 186678, 1063591, 6418167, 40860485, 273513831, 1919284246, 14080876273, 107750778177, 858195666410, 7100543662976, 60922480229704, 541193416875432, 4970306167860426
Offset: 0
-
{a(n)=if(n<0,0,if(n<=1,1,sum(k=0,n\2+1,(binomial(n-k,k)+binomial(n-k+1,k-1))*a(n-k-1))))}
A127785
G.f. satisfies: A(x) = 1 + x*A(x+x^2)^3.
Original entry on oeis.org
1, 1, 3, 15, 88, 594, 4404, 35260, 300918, 2714187, 25715364, 254752536, 2629587927, 28202629918, 313575278100, 3607684547463, 42880301468830, 525812475574788, 6643855672178904, 86407881067891029, 1155583581407910414
Offset: 0
-
{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^3,x,x+x^2));polcoeff(A,n)}
A140046
G.f. satisfies: A(x) = x/(1 - A(x+x^2)).
Original entry on oeis.org
1, 1, 3, 10, 41, 186, 922, 4911, 27830, 166656, 1049410, 6922476, 47698148, 342483885, 2557538781, 19829608532, 159393394129, 1326509171669, 11415703608635, 101473987987073, 930688926616454, 8798656042121634
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 41*x^5 + 186*x^6 + 922*x^7 +...
A(x+x^2) = x + 2*x^2 + 5*x^3 + 20*x^4 + 90*x^5 + 454*x^6 + 2488*x^7 +...
Let B(x) = x + x^2; define B_{n+1}(x) = B( B_{n}(x) ) with B_0(x)=x;
then g.f. A(x) equals the continued fraction:
A(x) = x/(1 - B(x)/(1 - B_2(x)/(1 - B_3(x)/(1 - B_4(x)/(1 - ...)))))
where B_{n}(x) begin:
B_2(x) = x + 2*x^2 + 2*x^3 + x^4 ;
B_3(x) = x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8 ;
B_4(x) = x + 4*x^2 + 12*x^3 + 30*x^4 + 64*x^5 + 118*x^6 + 188*x^7 +...;
B_5(x) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + 630*x^6 + 1656*x^7 +...
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{a(n)=local(A=x);if(n==0,A=x,for(i=1,n,A=x/(1-subst(A,x,x+x^2 +x*O(x^n))))); polcoeff(A,n)}
Showing 1-9 of 9 results.
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