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 1-6 of 6 results.

A030527 A convolution triangle of numbers obtained from A036083.

Original entry on oeis.org

1, 15, 1, 175, 30, 1, 1750, 575, 45, 1, 15750, 8750, 1200, 60, 1, 131250, 114625, 24375, 2050, 75, 1, 1031250, 1347500, 414750, 52000, 3125, 90, 1, 7734375, 14575000, 6208125, 1084875, 95000, 4425, 105, 1, 55859375, 147468750, 84184375
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,m) := s1p(6; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). Signed version: (-1)^(n-m)*a(n,m) := s1(6; n,m).

Examples

			{1}; {15,1}; {175,30,1}; {1750,575,45,1}; {15750,8750,1200,60,1}; ...

Links

Crossrefs

a(n, 1)= A036083(n-1). Row sums = A046088(n).

Formula

a(n, m) = 5*(5*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

A033842 Triangle of coefficients of certain polynomials (exponents in decreasing order).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 16, 16, 6, 1, 125, 125, 50, 10, 1, 1296, 1296, 540, 120, 15, 1, 16807, 16807, 7203, 1715, 245, 21, 1, 262144, 262144, 114688, 28672, 4480, 448, 28, 1, 4782969, 4782969, 2125764, 551124, 91854, 10206, 756, 36, 1, 100000000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

See A049323.

Examples

			{1}; {1,1}; {3,3,1}; {16,16,6,1}; {125,125,50,10,1}; .... E.g. third row {3,3,1} corresponds to polynomial p(2,x)= 3*x^2+3*x+1.

Links

  • W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
  • Thierry Lévy, The Number of Prefixes of Minimal Factorisations of a Cycle, The Electronic Journal of Combinatorics, 23(3) (2016), #P3.35

Crossrefs

a(n, 0)= A000272(n+1), n >= 0 (first column), a(n, 1)= A000272(n+1), n >= 1 (second column). p(k-1, -x)/(1-k*x)^k = (-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
See also A049323.

Formula

a(n, m) = binomial(n+1, m)*(n+1)^(n-m-1), n >= m >= 0 else 0.

A053110 Expansion of (-1 + 1/(1-7*x)^7)/(49*x); related to A036226.

Original entry on oeis.org

1, 28, 588, 10290, 158466, 2218524, 28840812, 353299947, 4121832715, 46164526408, 499416240232, 5243870522436, 53648829191076, 536488291910760, 5257585260725448, 50604258134482437, 479252091744216021
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A036226, A001792, A036083, A036224.

Programs

  • Magma
    [7^(n-1)*Binomial(n+7, 6): n in [0..30]]; // G. C. Greubel, Aug 16 2018
  • Mathematica
    CoefficientList[Series[(-1+1/(1-7x)^7)/(49x),{x,0,30}],x] (* or *) LinearRecurrence[{49,-1029,12005,-84035,352947,-823543,823543},{1,28,588,10290,158466,2218524,28840812},30] (* Harvey P. Dale, Jun 03 2015 *)
Table[7^(n-1)*Binomial[n+7, 6], {n,0,30}] (* G. C. Greubel, Aug 16 2018 *)
  • PARI
    vector(30,n,n--; 7^(n-1)*binomial(n+7, 6)) \\ G. C. Greubel, Aug 16 2018
  • Sage
    [lucas_number2(n, 7, 0)*binomial(n,6)/7^8 for n in range(7, 24)] # Zerinvary Lajos, Mar 13 2009

Formula

a(n) = 7^(n-1)*binomial(n+7, 6);
G.f.: (-1 + (1-7*x)^(-7))/(x*7^2).

A053113 Expansion of (-1 + 1/(1-10*x)^10)/(100*x); related to A053109.

Original entry on oeis.org

1, 55, 2200, 71500, 2002000, 50050000, 1144000000, 24310000000, 486200000000, 9237800000000, 167960000000000, 2939300000000000, 49742000000000000, 817190000000000000, 13075040000000000000, 204297500000000000000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

This is the tenth member of the k-family of sequences a(k,n) := k^(n-1)*binomial(n+k,k-1) starting with A000012 (powers of 1), A001792, A036068, A036070, A036083, A036224, A053110-113 for k=1..10.

Links

Programs

  • Magma
    [10^(n-1)*Binomial(n+10, 9): n in [0..30]]; // G. C. Greubel, Aug 16 2018
  • Mathematica
    Table[10^(n - 1)*Binomial[n + 10, 9], {n, 0, 30}] (* G. C. Greubel, Aug 16 2018 *)
LinearRecurrence[{100,-4500,120000,-2100000,25200000,-210000000,1200000000,-4500000000,10000000000,-10000000000},{1,55,2200,71500,2002000,50050000,1144000000,24310000000,486200000000,9237800000000},20] (* Harvey P. Dale, Jul 30 2025 *)
  • PARI
    vector(30,n,n--; 10^(n-1)*binomial(n+10, 9)) \\ G. C. Greubel, Aug 16 2018

Formula

a(n) = 10^(n-1)*binomial(n+10, 9).
G.f.: (-1 + (1-10*x)^(-10))/(x*10^2).

A036224 Expansion of (-1+1/(1-6*x)^6)/(36*x); related to A036084.

Original entry on oeis.org

1, 21, 336, 4536, 54432, 598752, 6158592, 60046272, 560431872, 5043886848, 44019376128, 374164697088, 3108445175808, 25311625003008, 202493000024064, 1594632375189504, 12381851383824384, 94927527275986944
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A036084, A036083.

Programs

  • Sage
    [lucas_number2(n, 6, 0)*binomial(n,5)/6^7for n in range(6, 24)] # Zerinvary Lajos, Mar 13 2009

Formula

a(n) = 6^(n-1)*binomial(n+6, 5);
g.f. (-1+(1-6*x)^(-6))/(x*6^2).

A049323 Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 50, 125, 125, 1, 15, 120, 540, 1296, 1296, 1, 21, 245, 1715, 7203, 16807, 16807, 1, 28, 448, 4480, 28672, 114688, 262144, 262144, 1, 36, 756, 10206, 91854, 551124, 2125764, 4782969, 4782969, 1, 45, 1200, 21000, 252000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

These polynomials p(n, x) appear in the W. Lang reference as c1(-(n+1);x), n >= 0 on p.12. The coefficients are given there in eq.(44) on p. 6. - Wolfdieter Lang, Nov 20 2015

Examples

			The triangle a(n, m) begins:
n\m 0  1   2    3     4      5      6      7 ...
0:  1
1:  1  1
2:  1  3   3
3:  1  6  16   16
4:  1 10  50  125  125
5:  1 15 120  540  1296  1296
6:  1 21 245 1715  7203  16807  16807
7:  1 28 448 4480 28672 114688 262144 262144
... reformatted. - Wolfdieter Lang, Nov 20 2015
E.g. the third row {1,3,3} corresponds to polynomial p(2,x)= 1 + 3*x + 3*x^2.

Links

Crossrefs

a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal), a(n, n-1)= A000272(n+1), n >= 1.
For n = 0..5 the row sequences a(n, m), m >= 0, are the first columns of the triangles A023531 (unit matrix), A030528, A049324, A049325, A049326, A049327, respectively.
Cf. A033842, A046757.

Programs

  • Magma
    /* As triangle: */ [[Binomial(n+1, k+1)*(n+1)^(k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 20 2015
  • Maple
    seq(seq(binomial(n+1,m+1)*(n+1)^(m-1),m=0..n),n=0..10); # Robert Israel, Oct 19 2015
  • Mathematica
    Table[Binomial[n + 1, k + 1] (n + 1)^(k - 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 19 2015 *)

Formula

a(n, m) = A033842(n, n-m) = binomial(n+1, m+1)*(n+1)^{m-1}, n >= m >= 0, else 0.
p(k-1, -x)/(1-k*x)^k =(-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
From Werner Schulte, Oct 19 2015: (Start)
a(2*n,n) = A000108(n)*(2*n+1)^n;
a(3*n,2*n) = A001764(n)*(3*n+1)^(2*n);
a(p*n,(p-1)*n) = binomial(p*n,n)/((p-1)*n+1)*(p*n+1)^((p-1)*n) for p > 0;
Sum_{m=0..n} (m+1)*a(n,m) = (n+2)^n;
Sum_{m=0..n} (-1)^m*(m+1)*a(n,m) = (-n)^n where 0^0 = 1;
p(n,x) = Sum_{m=0..n} a(n,m)*x^m = ((1+(n+1)*x)^(n+1)-1)/((n+1)^2*x).
(End)
Showing 1-6 of 6 results.

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