A131338
Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 29, 43, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 37, 51, 71, 100, 143, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 35, 46, 61, 81, 108, 145, 196
Offset: 0
Triangle begins:
1;
1, 1;
1,1, 1,2;
1,1,1, 1,2,3,5;
1,1,1,1, 1,2,3,4,6,9,14;
1,1,1,1,1, 1,2,3,4,5,7,10,14,20,29,43;
1,1,1,1,1,1, 1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,143;
1,1,1,1,1,1,1, 1,2,3,4,5,6,7,9,12,16,21,27,35,46,61,81,108,145,196,267,367,510; ...
Row sums equal the row sums (A098569) of triangle A098568,
where A098568(n, k) = binomial( (k+1)*(k+2)/2 + n-k-1, n-k):
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 21, 10, 1;
1, 15, 56, 55, 15, 1;
1, 21, 126, 220, 120, 21, 1; ...
-
T(n,k)=if(k>n*(n+1)/2 || k<0,0,if(k<=n,1,sum(i=0,k-n,T(n-1,i))))
for(n=0, 10, for(k=0, n*(n+1)/2, print1(T(n, k), ", ")); print(""))
A157133
G.f. satisfies: A(x) = Sum_{n>=0} x^(n(n+1)/2) * A(x)^n.
Original entry on oeis.org
1, 1, 1, 2, 4, 7, 14, 30, 62, 129, 278, 604, 1313, 2883, 6386, 14203, 31733, 71272, 160725, 363670, 825653, 1880351, 4293985, 9830499, 22558939, 51880565, 119552907, 276012657, 638348123, 1478749229, 3430799333, 7971134523
Offset: 0
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 14*x^6 + 30*x^7 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 13*x^4 + 26*x^5 + 54*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 30*x^4 + 66*x^5 + 145*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 59*x^4 + 140*x^5 + 326*x^6 +...
where
A(x) = 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 +...
-
{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,(A=sum(m=0,sqrtint(2*n+1),x^(m*(m+1)/2)*A^m)));polcoeff(A,n)}
A326424
G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)/2) * x^n = Sum_{n>=0} (1+x)^(n*(n-1)/2) * x^n.
Original entry on oeis.org
1, 0, 1, 0, 3, 4, 20, 62, 251, 1002, 4295, 19086, 88369, 423957, 2104214, 10783054, 56969183, 309900293, 1733790827, 9965992962, 58801256594, 355808106682, 2206237014216, 14007443494601, 90994768741426, 604395083728629, 4101881493676885, 28426771732773415, 201044377117957190, 1450195412613951590, 10663346917944740350, 79885242459500736025
Offset: 0
G.f.: A(x) = 1 + x^2 + 3*x^4 + 4*x^5 + 20*x^6 + 62*x^7 + 251*x^8 + 1002*x^9 + 4295*x^10 + 19086*x^11 + 88369*x^12 + 423957*x^13 + 2104214*x^14 + ...
such that the following series are equal
B(x) = 1 + A(x)*x + A(x)^3*x^2 + A(x)^6*x^3 + A(x)^10*x^4 + A(x)^15*x^5 + A(x)^21*x^6 + A(x)^28*x^7 + A(x)^36*x^8 + A(x)^45*x^9 + ...
and
B(x) = 1 + x + (1+x)*x^2 + (1+x)^3*x^3 + (1+x)^6*x^4 + (1+x)^10*x^5 + (1+x)^15*x^6 + (1+x)^21*x^7 + (1+x)^28*x^8 + (1+x)^36*x^9 + ...
where
B(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 27*x^6 + 81*x^7 + 262*x^8 + 910*x^9 + 3363*x^10 + 13150*x^11 + 54135*x^12 + ... + A121690(n-1)*x^n + ...
-
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A]=polcoeff( sum(m=0,#A, x^m*(1+x +x*O(x^#A))^(m*(m-1)/2) - x^m*Ser(A)^(m*(m+1)/2) ),#A)); A[n+1]}
for(n=0,35,print1(a(n),", "))
A325298
G.f. A(x) satisfies: Sum_{n>=0} x^(n*(n+1)/2) * A(x)^n = Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2).
Original entry on oeis.org
1, 2, 3, 6, 17, 56, 189, 673, 2561, 10321, 43612, 192439, 884702, 4227202, 20942697, 107363291, 568547892, 3105231155, 17467413871, 101069173004, 600841031279, 3665958252167, 22933712331957, 146968161483626, 963973640814332, 6466300466801210, 44327544752355141, 310325239786656220, 2217191324979383686, 16157187739844358535, 120020165206009363396, 908305634422244782653, 6999639387956913535113
Offset: 0
G.f.: A(x) = 1 + 2*x + 3*x^2 + 6*x^3 + 17*x^4 + 56*x^5 + 189*x^6 + 673*x^7 + 2561*x^8 + 10321*x^9 + 43612*x^10 + 192439*x^11 + 884702*x^12 + ...
such that the following series are equal
B(x) = 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 + x^15*A(x)^5 + x^21*A(x)^6 + x^28*A(x)^7 + x^36*A(x)^8 + x^45*A(x)^9 + ...
B(x) = 1 + x*(1+x) + x^2*(1+x)^3 + x^3*(1+x)^6 + x^4*(1+x)^10 + x^5*(1+x)^15 + x^6*(1+x)^21 + x^7*(1+x)^28 + x^8(1+x)^36 + x^9*(1+x)^45 + ...
where
B(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 81*x^6 + 262*x^7 + 910*x^8 + 3363*x^9 + 13150*x^10 + 54135*x^11 + 233671*x^12 + ... + A121690(n)*x^n + ...
-
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = -polcoeff( sum(m=0,#A, x^(m*(m+1)/2)*Ser(A)^m - x^m*(1+x +x*O(x^#A) )^(m*(m+1)/2) ),#A) );A[n+1]}
for(n=0,35,print1(a(n),", "))
A384832
G.f. A(x) = Sum_{n>=0} x^n * Product_{k=0..n} ((1+x)^(n-k+1) - x^k).
Original entry on oeis.org
1, 2, 4, 13, 41, 144, 533, 2072, 8463, 36142, 160852, 744491, 3576342, 17796825, 91587499, 486686277, 2666612930, 15045088274, 87301643726, 520416443472, 3183640482658, 19967208261651, 128273336978302, 843360769602607, 5670286993205471, 38957428760628861, 273318099568893757, 1956848333035887861
Offset: 1
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 13*x^4 + 41*x^5 + 144*x^6 + 533*x^7 + 2072*x^8 + 8463*x^9 + 36142*x^10 + 160852*x^11 + 744491*x^12 + ...
where
A(x) = 1 * ((1+x) - 1) +
x * ((1+x)^2 - 1)*((1+x) - x) +
x^2 * ((1+x)^3 - 1)*((1+x)^2 - x)*((1+x) - x^2) +
x^3 * ((1+x)^4 - 1)*((1+x)^3 - x)*((1+x)^2 - x^2)*((1+x) - x^3) +
x^4 * ((1+x)^5 - 1)*((1+x)^4 - x)*((1+x)^3 - x^2)*((1+x)^2 - x^3)*((1+x) - x^4) +
x^5 * ((1+x)^6 - 1)*((1+x)^5 - x)*((1+x)^4 - x^2)*((1+x)^3 - x^3)*((1+x)^2 - x^4)*((1+x) - x^5) +
x^6 * ((1+x)^7 - 1)*((1+x)^6 - x)*((1+x)^5 - x^2)*((1+x)^4 - x^3)*((1+x)^3 - x^4)*((1+x)^2 - x^5)*((1+x) - x^6) + ...
equivalently,
A(x) = x +
(2*x^2 + x^3) +
(3*x^3 + 9*x^4 + 10*x^5 + 5*x^6 - 2*x^7 - 3*x^8 - x^9) +
(4*x^4 + 26*x^5 + 78*x^6 + 139*x^7 + 147*x^8 + 73*x^9 - 25*x^10 - 65*x^11 - 45*x^12 - 15*x^13 - 2*x^14) +
(5*x^5 + 55*x^6 + 290*x^7 + 965*x^8 + 2226*x^9 + 3689*x^10 + 4378*x^11 + 3463*x^12 + 1184*x^13 - 1161*x^14 - 2296*x^15 - 2002*x^16 - 1034*x^17 - 239*x^18 + 85*x^19 + 102*x^20 + 44*x^21 + 10*x^22 + x^23) +
(6*x^6 + 99*x^7 + 794*x^8 + 4099*x^9 + 15185*x^10 + 42667*x^11 + 93837*x^12 + 164301*x^13 + 229972*x^14 + 253682*x^15 + 208380*x^16 + 100483*x^17 - 28293*x^18 - 125093*x^19 - 157729*x^20 - 130285*x^21 - 73656*x^22 - 21858*x^23 + 7068*x^24 + 14241*x^25 + 10381*x^26 + 4903*x^27 + 1605*x^28 + 355*x^29 + 48*x^30 + 3*x^31) + ...
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{a(n) = my(A = sum(m=0,n, x^m * prod(k=0,m, (1+x)^(m-k+1) - x^k +x*O(x^n)) )); polcoef(A,n)}
for(n=1,30,print1(a(n),", "))
Showing 1-5 of 5 results.