A129357
G.f.: A(x) = Product_{n>=1} [ (1-x)^4*(1 + 4x + 10x^2 +...+ n(n+1)(n+2)/3!*x^(n-1)) ].
Original entry on oeis.org
1, -4, -4, 36, -64, 256, -1328, 4488, -11406, 17700, 14716, -194508, 662768, -1374476, 2210780, -5820284, 25965483, -95963664, 259794360, -545959440, 952758316, -1278120568, 60070208, 8030404744, -34554134770, 94549651780, -196087124052, 330754522268, -511020392180
Offset: 0
G.f.: A(x) = (1-4x+6x^2-4x^3+x^4)*(1-10x^2+20x^3-15x^4+4x^5)*(1-20x^3+45x^4-36x^5+10x^6)*(1-35x^4+84x^5-70x^6+20x^7)*...
Terms are divisible by 4 except at positions given by:
a(n) == 1 (mod 4) at n = 16*[0, 2, 5, 15, 22, 26, 40,...];
a(n) == -1 (mod 4) at n = 16*[1, 7, 12, 35, 51, 57,...];
a(n) == 2 (mod 4) at n = 8*[1,3,5,6,8,9,13,15,16,19,..,A129359(k),..].
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{a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-x)^4*sum(j=1,k,binomial(j+2,3)*x^(j-1)) +x*O(x^n)),n))}
A129356
G.f.: A(x) = Product_{n>=1} [ (1-x)^3*(1 + 3x + 6x^2 +...+ n(n+1)/2*x^(n-1)) ].
Original entry on oeis.org
1, -3, -3, 15, -15, 66, -261, 618, -1155, 1040, 2361, -11616, 23733, -27027, 29394, -132318, 545790, -1383459, 2418896, -3383679, 4278462, -3127320, -8332866, 42021990, -99069516, 160683318, -200247795, 214883010, -345461022, 1184850729, -3966311448, 9899287254, -18787986009
Offset: 0
A(x) = (1-3x+3x^2-x^3)(1-6x^2+8x^3-3x^4)(1-10x^3+15x^4-6x^5)*...
*( 1 - (n+1)(n+2)/2*x^n + n(n+2)*x^(n+1) - n(n+1)/2*x^(n+2) )*...
Terms are divisible by 3 except at positions given by:
a(n) == 1 (mod 3) at n = [0, 45, 63, 198, 234, 459,...,9*A036498(k),..];
a(n) == -1 (mod 3) at n = [9, 18, 108, 135, 315, 360,..,9*A036499(k),..].
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{a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-x)^3*sum(j=1,k,j*(j+1)/2*x^(j-1)) +x*O(x^n)),n))}
A129358
G.f.: A(x) = Product_{n>=1} [ (1-x)^5*(1 + 5x + 15x^2 +...+ n(n+1)(n+2)(n+3)/4!*x^(n-1)) ].
Original entry on oeis.org
1, -5, -5, 70, -180, 770, -4760, 20840, -68085, 147890, -795, -1679855, 8378195, -25065005, 56439545, -145200415, 612604910, -2764023765, 10020060660, -28723695265, 67618167310, -128945409045, 137921330680, 375948665405, -3167538981120, 12823443150644, -38103903888575
Offset: 0
A(x) = (1-5x+10x^2-10x^3+5x^4-x^5)*(1-15x^2+40x^3-45x^4+24x^5-5x^6)*(1-35x^3+105x^4-126x^5+70x^6-15x^7)*(1-70x^4+224x^5-280x^6+160x^7-35x^8)*...
Terms are divisible by 5 except at positions given by 25*A001318(n):
a(n) == 1 (mod 5) at n = [0, 125, 175, 550, 650,...,25*A036498(k),...];
a(n) == -1 (mod 5) at n = [25, 50, 300, 375, 875,...,25*A036499(k),...].
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{a(n)=if(n==0,1,polcoeff(prod(k=1,n,(1-x)^5*sum(j=1,k,binomial(j+3,4)*x^(j-1)) +x*O(x^n)),n))}
Showing 1-3 of 3 results.
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