A002940 -id:A002940 - cp's OEIS frontend

A134400 M * A007318, where M = triangle with (1, 1, 2, 3, ...) in the main diagonal and the rest zeros.

1, 1, 1, 2, 4, 2, 3, 9, 9, 3, 4, 16, 24, 16, 4, 5, 25, 50, 50, 25, 5, 6, 36, 90, 120, 90, 36, 6, 7, 49, 147, 245, 245, 147, 49, 7, 8, 64, 224, 448, 560, 448, 224, 64, 8, 9, 81, 324, 756, 1134, 1134, 756, 324, 81, 9, 10, 100, 450, 1200, 2100, 2520, 2100, 1200, 450, 100, 10

Offset: 0

Gary W. Adamson, Oct 23 2007

Row sums = A134401: (1, 2, 8, 24, 64, 160, 384, ...).
Triangle T(n,k), read by rows, given by [1,1,-1,1,0,0,0,0,0,...] DELTA [1,1,-1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. A134402*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 26 2007
For n > 0, from n athletes, select a team of k players and then choose a coach who is allowed to be on the team or not. - Geoffrey Critzer, Mar 13 2010
Row sums are A036289 if first term changed to zero. Diagonal sums are A023610, starting with the 2nd diagonal. Partial sums of diagonals are A002940 if first term changed to zero. - John Molokach, Jul 06 2013
For n > 0, T(n,k) is the number of states in Sokoban puzzle with n non-obstacles cells and k boxes (see Russell and Norvig at page 157). - Stefano Spezia, Dec 03 2023

			First few rows of the triangle:
  1;
  1,  1;
  2,  4,   2;
  3,  9,   9,   3;
  4, 16,  24,  16,   4;
  5, 25,  50,  50,  25,   5;
  6, 36,  90, 120,  90,  36,  6;
  7, 49, 147, 245, 245, 147, 49, 7;
  ...

Stuart Russell and Peter Norvig, Artificial Intelligence: A Modern Approach, Fourth Edition, Hoboken: Pearson, 2021.

Wikipedia, Sokoban

Cf. A002940, A003506, A007318, A036289, A134401, A134402.

T(2n,n) give A002011(n-1) for n>=1.

with(combstruct): for n from 0 to 10 do seq(`if`(n=0, 1, n)* count( Combination(n), size=m), m=0..n) od; # Zerinvary Lajos, Apr 09 2008

Join[{1},Table[Table[n*Binomial[n, k], {k,0, n}], {n, 10}]] //Flatten (* Geoffrey Critzer, Mar 13 2010 adapted by Stefano Spezia, Dec 03 2023 *)

From Geoffrey Critzer, Mar 13 2010: (Start)
T(0,0) = 1 and T(n,k) = n * binomial(n,k) for n > 0.
E.g.f. for column k is: (x^k/k!)*exp(x)*(x+k). (End)
T(n,k) = A003506(n,k) + A003506(n,k-1). - Geoffrey Critzer, Mar 13 2010
G.f.: (1-x-x*y+x^2+x^2*y+x^2*y^2)/(1-2*x-2*x*y+x^2+2*x^2*y+x^2*y^2). - Philippe Deléham, Nov 14 2013
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-2,k-2), T(0,0)=T(1,0)=T(1,1)=1, T(2,0)=T(2,2)=2, T(2,1)=4, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 14 2013
E.g.f.: 1 + exp(y*x)*exp(x)*(y*x + x). - Geoffrey Critzer, Mar 15 2015

a(55)-a(65) from Stefano Spezia, Dec 03 2023

A062124 Fourth column of A046741.

Original entry on oeis.org

3, 26, 94, 234, 473, 838, 1356, 2054, 2959, 4098, 5498, 7186, 9189, 11534, 14248, 17358, 20891, 24874, 29334, 34298, 39793, 45846, 52484, 59734, 67623, 76178, 85426, 95394, 106109, 117598, 129888, 143006, 156979, 171834, 187598, 204298

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (6+19*n+18*n^2+9*n^3)/2); # G. C. Greubel, Jan 31 2019

[(6+19*n+18*n^2+9*n^3)/2: n in [0..40]]; // G. C. Greubel, Jan 31 2019

Table[(6+19*n+18*n^2+9*n^3)/2, {n,0,40}] (* G. C. Greubel, Jan 31 2019 *)
LinearRecurrence[{4,-6,4,-1},{3,26,94,234},40] (* Harvey P. Dale, Feb 20 2022 *)

vector(40, n, n--; (6+19*n+18*n^2+9*n^3)/2) \\ G. C. Greubel, Jan 31 2019

[(6+19*n+18*n^2+9*n^3)/2 for n in range(40)] # G. C. Greubel, Jan 31 2019

G.f.: (3 + 14*x + 8*x^2 + 2*x^3)/(1-x)^4. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y) - (1+y)*x).
From G. C. Greubel, Jan 31 2019: (Start)
a(n) = (6 + 19*n + 18*n^2 + 9*n^3)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (6 + 46*x + 45*x^2 + 9*x^3)*exp(x)/2. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A202874 Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 5, 8, 8, 5, 8, 13, 14, 13, 8, 13, 21, 23, 23, 21, 13, 21, 34, 37, 39, 37, 34, 21, 34, 55, 60, 63, 63, 60, 55, 34, 55, 89, 97, 102, 103, 102, 97, 89, 55, 89, 144, 157, 165, 167, 167, 165, 157, 144, 89, 144, 233, 254, 267, 270, 272, 270, 267, 254

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=(1,2,3,5,8,13,...)=(F(k+1)), where F=A000045, and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202874 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202875 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....2....3....5....8....13
2....5....8....13...21...34
3....8....14...23...37...60
5....13...23...39...63...102
8....21...37...63...102..167

Cf. A202875.

s[k_] := Fibonacci[k + 1];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A001911 *)
Table[m[1, j], {j, 1, 12}]     (* A000045 *)
Table[m[j, j], {j, 1, 12}]     (* A119996 *)
Table[m[j, j + 1], {j, 1, 12}] (* A180664 *)
Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}]  (* A002940 *)

A062125 Fifth column of A046741.

Original entry on oeis.org

5, 56, 263, 815, 1982, 4115, 7646, 13088, 21035, 32162, 47225, 67061, 92588, 124805, 164792, 213710, 272801, 343388, 426875, 524747, 638570, 769991, 920738, 1092620, 1287527, 1507430, 1754381, 2030513, 2338040, 2679257, 3056540

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (40+126*n+165*n^2+90*n^3+27*n^4)/8); # G. C. Greubel, Jan 31 2019

[(40+126*n+165*n^2+90*n^3+27*n^4)/8: n in [0..40]]; // G. C. Greubel, Jan 31 2019

LinearRecurrence[{5, -10, 10, -5, 1}, {5, 56, 263, 815, 1982}, 31] (* or *) CoefficientList[Series[(5+33x^2+10x^3+31x+2x^4)/(1-x)^5,{x,0,30}],x] (* Harvey P. Dale, Dec 21 2011 *)
Table[(40+126*n+165*n^2+90*n^3+27*n^4)/8, {n,0,40}] (* G. C. Greubel, Jan 31 2019 *)

vector(40, n, n--; (40+126*n+165*n^2+90*n^3+27*n^4)/8) \\ G. C. Greubel, Jan 31 2019

[(40+126*n+165*n^2+90*n^3+27*n^4)/8 for n in range(40)] # G. C. Greubel, Jan 31 2019

G.f.: (5 + 33*x^2 + 10*x^3 + 31*x + 2*x^4)/(1-x)^5. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), where a(0)=5, a(1)=56, a(2)=263, a(3)=815, a(4)=1982. - Harvey P. Dale, Dec 21 2011
From G. C. Greubel, Jan 31 2019: (Start)
a(n) = (40 + 126*n + 165*n^2 + 90*n^3 + 27*n^4)/8.
E.g.f.: (40 + 408*x + 624*x^2 + 252*x^3 + 27*x^4)*exp(x)/8. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A062126 Sixth column of A046741.

Original entry on oeis.org

8, 114, 667, 2504, 7191, 17266, 36482, 70050, 124882, 209834, 335949, 516700, 768233, 1109610, 1563052, 2154182, 2912268, 3870466, 5066063, 6540720, 8340715, 10517186, 13126374, 16229866, 19894838, 24194298, 29207329

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5 )/40); # G. C. Greubel, Jan 31 2019

[(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40: n in [0..40]]; // G. C. Greubel, Jan 31 2019

Table[(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40, {n, 0, 40}] (* G. C. Greubel, Jan 31 2019 *)

vector(40, n, n--; (320+1114*n+1515*n^2+1125*n^3+405*n^4 + 81*n^5)/40) \\ G. C. Greubel, Jan 31 2019

[(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40 for n in range(40)] # G. C. Greubel, Jan 31 2019

G.f.: (x+2)*(2*x^4 + 8*x^3 + 36*x^2 + 31*x + 4)/(1-x)^6. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1 - y - y^2)*(1-y) - (1+y)*x).
From G. C. Greubel, Jan 31 2019: (Start)
a(n) = (320 + 1114*n + 1515*n^2 + 1125*n^3 + 405*n^4 + 81*n^5)/40.
E.g.f.: (320 + 4240*x + 8940*x^2 + 5580*x^3 + 1215*x^4 + 81*x^5)*exp(x)/40. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A308244 Triangle T(n,k) read by rows, giving even-numbered coefficients of the matching polynomial of the n-ladder graph.

Original entry on oeis.org

1, -1, 1, 2, -4, 1, -3, 11, -7, 1, 5, -26, 29, -10, 1, -8, 56, -94, 56, -13, 1, 13, -114, 263, -234, 92, -16, 1, -21, 223, -667, 815, -473, 137, -19, 1, 34, -424, 1577, -2504, 1982, -838, 191, -22, 1, -55, 789, -3538, 7018, -7191, 4115, -1356, 254, -25, 1, 89, -1444, 7622, -18336, 23431, -17266

Offset: 0

Robert Israel, May 16 2019

For 0 <= k <= n, T(n,k) is the coefficient of x^(2*k) in the matching polynomial of the n-ladder graph. We take T(0,0)=1.

			Triangle begins
1
-1 1
2 -4 1
-3 11 -7 1
5 -26 29 -10 1
-8 56 -94 56 -13 1
13 -114 263 -234 92 -16 1
-21 223 -667 815 -473 137 -19 1
34 -424 1577 -2504 1982 -838 191 -22 1

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
Eric Weisstein's World of Mathematics, Ladder Graph.
Eric Weisstein's World of Mathematics, Matching Polynomial.

Cf. A000045, A002940, A002941.

g:= gfun:-rectoproc({a(n+3)+(-x^2+2)*a(n+2)+x^2*a(n+1)-a(n),a(0)=1,a(1)=x^2-1,a(2)=x^4-4*x^2+2}, a(n), remember):
for nn from 0 to 10 do
seq(coeff(g(nn),x,k),k=0..2*nn,2)
od;

Generating function as triangle: (1+x)/(1+2*x-x*y+x^2*y-x^3).
T(n,k) = T(n-1,k-1)-2*T(n-1,k)-T(n-2,k-1)+T(n-3,k) for n >= 3 (taking T(n,k)=0 unless 0 <= k <= n).
T(n,0) = (-1)^n*A000045(n+1).
T(n,1) = (-1)^(n+1)*A002940(n) for n >= 1.
T(n,2) = (-1)^n*A002941(n-1) for n >= 2.

cp's OEIS Frontend

A134400 M * A007318, where M = triangle with (1, 1, 2, 3, ...) in the main diagonal and the rest zeros.

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A062124 Fourth column of A046741.

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A202874 Symmetric matrix based on (1,2,3,5,8,13,...), by antidiagonals.

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A062125 Fifth column of A046741.

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A062126 Sixth column of A046741.

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GAP

Magma

Mathematica

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Sage

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A308244 Triangle T(n,k) read by rows, giving even-numbered coefficients of the matching polynomial of the n-ladder graph.

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Keywords

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Programs

Maple

Formula