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 1-10 of 11 results. Next

A107719 Matrix cube-root of triangle A107717.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 43, 13, 3, 1, 509, 131, 25, 4, 1, 7579, 1741, 303, 41, 5, 1, 135341, 28451, 4681, 587, 61, 6, 1, 2813851, 549757, 87039, 10449, 1011, 85, 7, 1, 66733853, 12247211, 1885177, 220023, 20445, 1603, 113, 8, 1, 1778159275, 308953453
Offset: 0

Views

Author

Paul D. Hanna, May 30 2005

Keywords

Comments

Column 0 is A107720. Column 1 is A107721. Matrix inverse is A107727.

Examples

			Triangle begins:
1;
1,1;
5,2,1;
43,13,3,1;
509,131,25,4,1;
7579,1741,303,41,5,1;
135341,28451,4681,587,61,6,1;
2813851,549757,87039,10449,1011,85,7,1; ...

Crossrefs

Cf. A107717, A107720, A107721, A107727.

Programs

  • PARI
    T(n,k)=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

A107726 Matrix inverse of triangle A107717, read by rows.

Original entry on oeis.org

1, -3, 1, -3, -6, 1, -21, -3, -9, 1, -219, -21, -3, -12, 1, -2973, -219, -21, -3, -15, 1, -49323, -2973, -219, -21, -3, -18, 1, -964173, -49323, -2973, -219, -21, -3, -21, 1, -21680571, -964173, -49323, -2973, -219, -21, -3, -24, 1, -551173053, -21680571, -964173, -49323, -2973, -219, -21, -3, -27, 1
Offset: 0

Views

Author

Paul D. Hanna, May 30 2005

Keywords

Comments

Except for initial terms, each column is the same and equals negative A107716 (inverse INVERT of triple factorials).

Examples

			Triangle begins:
1;
-3,1;
-3,-6,1;
-21,-3,-9,1;
-219,-21,-3,-12,1;
-2973,-219,-21,-3,-15,1;
-49323,-2973,-219,-21,-3,-18,1;
-964173,-49323,-2973,-219,-21,-3,-21,1; ...
Matrix inverse is A107717:
1;
3,1;
21,6,1;
219,57,9,1;
2973,723,111,12,1;
49323,11361,1713,183,15,1; ...

Crossrefs

Cf. A107716, A107717.

Programs

  • PARI
    T(n,k)=if(n=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)))))[n+1,k+1])

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
Offset: 0

Views

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

  • 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])}

A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2*p-1)/2)) = (2*p-1)*(column p of T), or [T^((2*p-1)/2)](m,0) = (2*p-1)*T(p+m,p+1) for all m>=1 and p>=0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 74, 26, 6, 1, 706, 226, 50, 8, 1, 8162, 2426, 522, 82, 10, 1, 110410, 30826, 6498, 1010, 122, 12, 1, 1708394, 451586, 93666, 14458, 1738, 170, 14, 1, 29752066, 7489426, 1532970, 235466, 28226, 2754, 226, 16, 1, 576037442
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2005

Keywords

Comments

Column 0 is A000698 (related to double factorials), offset 1. Column 1 is A105616 (column 0 of T^(1/2), offset 1). The matrix logarithm divided by 2 yields the integer triangle A105629.
Compare with triangular matrix A107717, which satisfies: SHIFT_LEFT(column 0 of A107717^((3*k-1)/3)) = (3*k-1)*(column k of A107717).

Examples

			SHIFT_LEFT(column 0 of T^(-1/2)) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^(1/2)) = 1*(column 1 of T);
SHIFT_LEFT(column 0 of T^(3/2)) = 3*(column 2 of T);
SHIFT_LEFT(column 0 of T^(5/2)) = 5*(column 3 of T).
Triangle begins:
1;
2,1;
10,4,1;
74,26,6,1;
706,226,50,8,1;
8162,2426,522,82,10,1;
110410,30826,6498,1010,122,12,1;
1708394,451586,93666,14458,1738,170,14,1;
29752066,7489426,1532970,235466,28226,2754,226,16,1; ...
Matrix square-root T^(1/2) is A105623 which begins:
1;
1,1;
4,2,1;
26,10,3,1;
226,74,19,4,1;
2426,706,167,31,5,1; ...
compare column 0 of T^(1/2) to column 1 of T;
also, column 1 of T^(1/2) equals column 0 of T.
Matrix inverse square-root T^(-1/2) is A105620 which begins:
1;
-1,1;
-2,-2,1;
-10,-4,-3,1;
-74,-20,-7,-4,1;
-706,-148,-39,-11,-5,1; ...
compare column 0 of T^(-1/2) to column 0 of T.
Matrix inverse T^-1 is A105619 which begins:
1;
-2,1;
-2,-4,1;
-10,-2,-6,1;
-74,-10,-2,-8,1;
-706,-74,-10,-2,-10,1;
-8162,-706,-74,-10,-2,-12,1; ...

Crossrefs

Cf. A000698 (column 0), A105616 (column 1), A105617 (column 2), A105618 (row sums), A105619 (T^-1), A105620 (T^(-1/2)), A105623 (T^(1/2)), A105627 (T^(3/2)), A105629 (matrix log).
Cf. A107717.

Programs

  • PARI
    {T(n,k) = if(n
  • PARI
    {T(n,k) = if(n=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))

Formula

T(n, k) = 2*(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) = A000698(n+1) for n>=0.

A107716 Inverse INVERT transform of triple factorial numbers (3*n-2)!!! (A007559).

Original entry on oeis.org

1, 3, 21, 219, 2973, 49323, 964173, 21680571, 551173053, 15633866379, 489583062381, 16780438408539, 624935780160285, 25131869565110571, 1085528359404039117, 50124679063548821499, 2464153823558024331645, 128500643820213560377803, 7085182933810282490250285
Offset: 0

Views

Author

Paul D. Hanna, May 23 2005

Keywords

Comments

Column 0 of triangle A107717.

Examples

			The triple factorials begin: {1,4,28,280,3640,58240,...}; thus the inverse INVERT transform of the triple factorials can be calculated by the g.f.s:
1/(1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3640*x^5 + 58240*x^6 +...) = (1 - x - 3*x^2 - 21*x^3 - 219*x^4 - 2973*x^5 - 49323*x^6 -...).

Links

Crossrefs

Cf. A007559, A000698, A107717.

Programs

  • Maple
    b:= proc(n) b(n):=  `if`(n=0, 1, b(n-1)*(3*n+1)) end:
a:= proc(n) a(n):= -`if`(n<0, 1, add(a(n-i-1)*b(i), i=0..n)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 23 2017
  • Mathematica
    m = 20; f3[n_] := Product[3k+1, {k, 0, n-1}]; A[x_] = 1-1/(1+Sum[f3[n] x^n, {n, 1, m}]); CoefficientList[A[x] + O[x]^m, x] // Rest (* Jean-François Alcover, May 01 2019 *)
  • PARI
    a(n)=polcoeff(1-(1+sum(k=1,n+1,prod(j=0,k-1,3*j+1)*x^k)+x^2*O(x^n))^-1,n+1)

Formula

G.f.: A(x) = 1 - 1/[1 + Sum_{n>=1} (3*n-2)!!! * x^n ] where (3*n-2)!!! = Product_{k=0..n-1} (3*k+1).
a(n) = Sum_{k, 0<=k<=n} A089949(n, k)*3^k . - Philippe Deléham, Aug 15 2005
G.f.: (1 - Q(0))/x where Q(k) = 1 - x*(3*k+1)/(1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: 1/x - 2 - 2/x/G(0), where G(k)= 1 + 1/(1 - x*(3*k+3)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
From Peter Bala, May 23 2017: (Start)
G.f. A(x) = 1/(1 + x - 4*x/(1 + 4*x - 7*x/(1 + 7*x - 10*x/(1 + 10*x - ...)))).
A(x) = 1/(1 + x - 4*x/(1 - 3*x/(1 - 7*x/(1 - 6*x/(1 - 10*x/(1 - 9*x - ...)))))). (End)

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
    
				
    • 
				
			

		

	

	
	
	


		
			

	

		A107722
		Matrix square of triangle A107721, read by rows.
	

	
Original entry on oeis.org

	

		

			1, 2, 1, 12, 4, 1, 114, 32, 6, 1, 1446, 364, 62, 8, 1, 22722, 5276, 854, 102, 10, 1, 424302, 92332, 14466, 1664, 152, 12, 1, 9168162, 1889012, 289222, 32536, 2874, 212, 14, 1, 224961774, 44207212, 6652522, 737472, 63886, 4564, 282, 16, 1, 6180211842
		

		
Offset: 0

	

	

		
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Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
Matrix (2/3)-power of triangle A107717. SHIFT_LEFT of column 0 = 2*(column 1 of A107717). Matrix inverse is A107728.

		

	

	
	
	

		
Examples

		
			Triangle begins:
1;
2,1;
12,4,1;
114,32,6,1;
1446,364,62,8,1;
22722,5276,854,102,10,1;
424302,92332,14466,1664,152,12,1;
9168162,1889012,289222,32536,2874,212,14,1; ...
		

	

	
	
	
	
	

		
Crossrefs

		

			
Cf. A107717, A107721, A107728.

		

	

	
	
	

		
Programs

		

			
    
				
				
  • 
				
    PARI
    
				
    
				
    • 
				
			

		

	

	
	
	


		
			

	

		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
		

		
Offset: 0

	

	

		
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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

		

			
    
				
				
  • 
				
    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
				
    • 
				
			

		

	

	
	
	


		
			

	

		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
		

		
Offset: 0

	

	

		
Views

		

			
		

	

	

		
Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	
	
	
	
	

		
Crossrefs

		

			
Cf. A107717, A107719.

		

	

	
	
	

		
Programs

		

			
    
				
				
  • 
				
    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])}
    
				
    • 
				
			

		

	

	
	
	


		
			

	

		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
		

		
Offset: 0

	

	

		
Views

		

			
		

	

	

		
Author

		

			Paul D. Hanna, May 30 2005
		

	

	

		
Keywords

		

			
		

	

	
	
	
	
	
	

		
Crossrefs

		

			
Cf. A107717, A107719, A107724.

		

	

	
	
	

		
Programs

		

			
    
				
				
  • 
				
    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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