cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A107721 Column 1 of triangle A107719.

Original entry on oeis.org

1, 2, 13, 131, 1741, 28451, 549757, 12247211, 308953453, 8706012827, 271093643293, 9245857326635, 342832527086797, 13733511532751099, 591127774090746493, 27209560375187822795, 1333804962202136755501
Offset: 0

Views

Author

Paul D. Hanna, May 30 2005

Keywords

Crossrefs

Cf. A107717, A107719.

Programs

  • PARI
    
				
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		A107727
		Matrix inverse of A107719.
	

	
Original entry on oeis.org

	

		

			1, -1, 1, -3, -2, 1, -21, -7, -3, 1, -219, -53, -13, -4, 1, -2973, -583, -115, -21, -5, 1, -49323, -8249, -1437, -217, -31, -6, 1, -964173, -141655, -22715, -3101, -369, -43, -7, 1, -21680571, -2853185, -430877, -55251, -5975, -581, -57, -8, 1, -551173053, -65887783, -9505707, -1168349, -119137
		

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
Matrix square is A107728. Matrix cube is A107726. Column 0 is negative A107716 shift right.

		

	

	
	
	

		
Examples

		
			Triangle begins:
1;
-1,1;
-3,-2,1;
-21,-7,-3,1;
-219,-53,-13,-4,1;
-2973,-583,-115,-21,-5,1;
-49323,-8249,-1437,-217,-31,-6,1;
-964173,-141655,-22715,-3101,-369,-43,-7,1; ...
		

	

	
	
	
	
	

		
Crossrefs

		

			
Cf. A107716, A107719, A107726, A107728.

		

	

	
	
	

		
Programs

		

			
    
				
				
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    PARI
    
				
    {T(n,k)=local(L,N,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, polcoeff(1/sum(i=0,m-j,prod(r=0,i-1,3*r+1)*x^i)+O(x^m),m-j)))))^-1); L=sum(i=1,#M,(M^0-M)^i/i)/3;N=sum(i=0,#L,L^i/i!); return(if(n<0,0,N[n+1,k+1]))}
    
				
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		A107724
		Matrix logarithm of triangle A107719, read by rows.
	

	
Original entry on oeis.org

	

		

			0, 1, 0, 4, 2, 0, 31, 10, 3, 0, 343, 88, 19, 4, 0, 4855, 1066, 199, 31, 5, 0, 83209, 16216, 2779, 382, 46, 6, 0, 1670743, 295360, 47791, 6130, 655, 64, 7, 0, 38436673, 6254824, 970849, 119182, 11929, 1036, 85, 8, 0, 996825703, 150917560, 22697647, 2703730
		

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
Also equals one-third of the matrix logarithm of A107717. Column 0 is A107725.

		

	

	
	
	

		
Examples

		
			Triangle begins:
0;
1,0;
4,2,0;
31,10,3,0;
343,88,19,4,0;
4855,1066,199,31,5,0;
83209,16216,2779,382,46,6,0;
1670743,295360,47791,6130,655,64,7,0;
38436673,6254824,970849,119182,11929,1036,85,8,0; ...
		

	

	
	
	
	
	

		
Crossrefs

		

			
Cf. A107717, A107719.

		

	

	
	
	

		
Programs

		

			
    
				
				
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    PARI
    
				
    T(n,k)=local(L,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, polcoeff(1/sum(i=0,m-j,prod(r=0,i-1,3*r+1)*x^i)+O(x^m),m-j)))))^-1); L=sum(i=1,#M,(-1)^(i-1)*(M-M^0)^i/i);return(if(n
				
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		A107720
		Column 0 of triangle A107719.
	

	
Original entry on oeis.org

	

		

			1, 1, 5, 43, 509, 7579, 135341, 2813851, 66733853, 1778159275, 52604224205, 1711276244731, 60729274013309, 2335153500391627, 96727188777453869, 4294441686826824091, 203456100846249179357, 10245810557884742785387
		

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	
	
	
	
	

		
Crossrefs

		

			
Cf. A107717, A107719.

		

	

	
	
	

		
Programs

		

			
    
				
				
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    PARI
    
				
    {a(n)=local(E,L,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, -polcoeff(1-(1+sum(c=1,m-j,prod(i=0,c-1,3*i+1)*x^c)+x*O(x^(m-j)))^-1,m-j);))))^-1); L=matrix(#M,#M,r,c,if(r>=c,sum(i=1,#M,(-1)^(i-1)*(M-M^0)^i/i)[r,c])); E=matrix(#L,#L,r,c,if(r>=c,sum(i=0,#L,L^i/3^i/i!)[r,c])); if(n<0,0,E[n+1,1])}
    
				
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		A107725
		Column 0 of A107724, which is the matrix logarithm of triangle A107719.
	

	
Original entry on oeis.org

	

		

			0, 1, 4, 31, 343, 4855, 83209, 1670743, 38436673, 996825703, 28778874481, 915636860023, 31837734460129, 1201360385000071, 48899006799783889, 2135779996980897175, 99646497213608842561, 4946194601237466540967
		

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	
	
	
	
	

		
Crossrefs

		

			
Cf. A107717, A107719, A107724.

		

	

	
	
	

		
Programs

		

			
    
				
				
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    PARI
    
				
    {a(n)=local(L,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, polcoeff(1/sum(i=0,m-j,prod(r=0,i-1,3*r+1)*x^i)+O(x^m),m-j)))))^-1); L=sum(i=1,#M,(-1)^(i-1)*(M-M^0)^i/i);return(if(n<0,0,L[n+1,1]/3))}
    
				
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		A107717
		Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((3*p-1)/3)) = (3*p-1)*(column p of T), or [T^((3*p-1)/3)](m,0) = (3*p-1)*T(p+m,p) for all m>=1 and p>=0.
	

	
Original entry on oeis.org

	

		

			1, 3, 1, 21, 6, 1, 219, 57, 9, 1, 2973, 723, 111, 12, 1, 49323, 11361, 1713, 183, 15, 1, 964173, 212151, 31575, 3351, 273, 18, 1, 21680571, 4584081, 675489, 71391, 5799, 381, 21, 1, 551173053, 112480887, 16442823, 1732881, 140529, 9219, 507, 24, 1
		

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
Column 0 is A107716 (INVERTi of triple factorials). Column 1 is A107718 (twcie column 0 of T^(2/3), offset 1). The matrix logarithm divided by 3 yields the integer triangle A107724.

		

	

	
	
	

		
Examples

		
			SHIFT_LEFT(column 0 of T^(p-1/3)) = (3*p-1)*(column p of T):
SHIFT_LEFT(column 0 of T^(-1/3)) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^(2/3)) = 2*(column 1 of T);
SHIFT_LEFT(column 0 of T^(5/3)) = 5*(column 2 of T).
Triangle begins:
1;
3,1;
21,6,1;
219,57,9,1;
2973,723,111,12,1;
49323,11361,1713,183,15,1;
964173,212151,31575,3351,273,18,1;
21680571,4584081,675489,71391,5799,381,21,1; ...
Matrix power (2/3), T^(2/3), is A107719 and begins:
1;
2,1;
12,4,1;
114,32,6,1;
1446,364,62,8,1;
22722,5276,854,102,10,1; ...
compare column 0 of T^(2/3) to 2*(column 1 of T).
Matrix inverse cube-root T^(-1/3) is A107727 and begins:
1;
-1,1;
-3,-2,1;
-21,-7,-3,1;
-219,-53,-13,-4,1;
-2973,-583,-115,-21,-5,1; ...
compare column 0 of T^(-1/3) to column 0 of T.
Matrix inverse is A107726 and begins:
1;
-3,1;
-3,-6,1;
-21,-3,-9,1;
-219,-21,-3,-12,1;
-2973,-219,-21,-3,-15,1; ...
compare column 0 of T^(-1) to column 0 of T.
		

	

	
	
	
	
	

		
Crossrefs

		

			
Cf. A105615, A107716, A107718-A107727.

		

	

	
	
	

		
Programs

		

			
    
				
				
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    PARI
    
				
    {T(n,k)=if(n
				
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    PARI
    
				
    {T(n,k)=if(n=j,if(m==j,1,if(m==j+1,-3*j,-T(m-j-1,0)))))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))
    
				
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Formula

		

			
T(n, k) = 3*(k+1)*T(n, k+1) + Sum_{j=1..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>=0, with T(n, n) = 1 for n>=0. T(n, 0) = A107716(n+1) for n>=0.

		

	

	
	


		
			

	

		A107728
		Matrix inverse of A107722.
	

	
Original entry on oeis.org

	

		

			1, -2, 1, -4, -4, 1, -26, -8, -6, 1, -262, -52, -14, -8, 1, -3482, -524, -102, -22, -10, 1, -56902, -6964, -1130, -184, -32, -12, 1, -1099514, -113804, -16326, -2304, -306, -44, -14, 1, -24494422, -2199028, -287882, -37224, -4326, -476, -58, -16, 1, -617906906, -48988844, -5969382, -727928, -78114
		

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
Column 0 shift left = -2*A107721, where A107721 = column 1 of A107719. Column 1 shift left = 2*(column 0) shift left. Matrix square of A107727.

		

	

	
	
	

		
Examples

		
			Triangle begins:
1;
-2,1;
-4,-4,1;
-26,-8,-6,1;
-262,-52,-14,-8,1;
-3482,-524,-102,-22,-10,1;
-56902,-6964,-1130,-184,-32,-12,1;
-1099514,-113804,-16326,-2304,-306,-44,-14,1; ...
		

	

	
	
	
	
	

		
Crossrefs

		

			
Cf. A107719, A107721, A107722, A107727.

		

	

	
	
	

		
Programs

		

			
    
				
				
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    PARI
    
				
    {T(n,k)=local(L,N,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, polcoeff(1/sum(i=0,m-j,prod(r=0,i-1,3*r+1)*x^i)+O(x^m),m-j)))))^-1); L=sum(i=1,#M,(M^0-M)^i/i)/3;N=sum(i=0,#L,L^i/i!); return(if(n<0,0,(N^2)[n+1,k+1]))}
    
				
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		A107718
		Column 1 of triangle A107717.
	

	
Original entry on oeis.org

	

		

			1, 6, 57, 723, 11361, 212151, 4584081, 112480887, 3090105921, 93988998183, 3136338148017, 113945190405303, 4477940877230625, 189296643095867847, 8565988634172222609, 413169192012610306263, 21161884092470464784385
		

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
Equals one-half of column 0 of A107719 shift 1 place left.

		

	

	
	
	
	
	
	

		
Crossrefs

		

			
Cf. A107717, A107719.

		

	

	
	
	

		
Programs

		

			
    
				
				
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    PARI
    
				
    {a(n)=if(n<0,0,(matrix(n+2,n+2,m,j,if(m>=j,if(m==j,1,if(m==j+1,-3*j, polcoeff(1/sum(i=0,m-j,prod(r=0,i-1,3*r+1)*x^i)+O(x^m),m-j)))))^-1)[n+2,2])}
    
				
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		Showing 1-8 of 8 results.
		
	


		

			
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