A117401 -id:A117401 - cp's OEIS frontend

A176644 Triangle T(n, k) = 40^(k*(n-k)), read by rows.

1, 1, 1, 1, 40, 1, 1, 1600, 1600, 1, 1, 64000, 2560000, 64000, 1, 1, 2560000, 4096000000, 4096000000, 2560000, 1, 1, 102400000, 6553600000000, 262144000000000, 6553600000000, 102400000, 1, 1, 4096000000, 10485760000000000, 16777216000000000000, 16777216000000000000, 10485760000000000, 4096000000, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,         1;
  1,        40,             1;
  1,      1600,          1600,               1;
  1,     64000,       2560000,           64000,             1;
  1,   2560000,    4096000000,      4096000000,       2560000,         1;
  1, 102400000, 6553600000000, 262144000000000, 6553600000000, 102400000, 1;

G. C. Greubel, Rows n = 0..40 of the triangle, flattened

Cf. A000567.
Cf. A176642 (q=2), A176643 (q=3), this sequence (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26), this sequence (m=38).

[40^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021

T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 4], {n, 0, 12}, {k, 0, n}]//Flatten
Table[(40)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)

flatten([[40^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 4.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 4.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 38. - G. C. Greubel, Jul 01 2021

Edited by G. C. Greubel, Jul 01 2021

A117403 a(n) = Sum_{k=0..floor(n/2)} 2^((n-2k)k) for n>=0.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 34, 105, 386, 1681, 8706, 53793, 395266, 3442753, 35659778, 440672385, 6476038146, 112812130561, 2336999211010, 57759810847233, 1697654543745026, 59146046307566593, 2450521284684021762

Offset: 0

Paul D. Hanna, Mar 12 2006

nonn

Equals the antidiagonal sums of triangle A117401.

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 13*x^5 + 34*x^6 + 105*x^7 + ...
where
A(x) = 1/(1-x^2) + x/(1-2*x^2) + x^2/(1-4*x^2) + x^3/(1-8*x^2) + x^4/(1-16*x^2) + ...

G. C. Greubel, Table of n, a(n) for n = 0..150

Cf. A117401 (triangle), A117402 (row sums).

[(&+[2^(k*(n-2*k)) : k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 28 2021

Table[Sum[2^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Jun 28 2021 *)

PARI
```
a(n) = sum(k=0,n\2,2^((n-2*k)*k))
```

{a(n) = polcoeff(sum(m=0,n,x^m/(1-2^m*x^2 +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

[sum(2^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # G. C. Greubel, Jun 28 2021

G.f.: A(x) = Sum_{n>=0} x^n / (1 - 2^n*x^2).
a(2*n) = Sum_{k=0..n} 4^((n-k)*k).
a(2*n+1) = Sum_{k=0..n} 2^k * 4^((n-k)*k).
G.f.: 1/(1-x^2) - x/(Q(0) +x-x^3), where Q(k) = x^2*(2+x)*2^k -1-x - x*(2*x^2*2^k -1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 11 2013

Name changed by Paul D. Hanna, Nov 11 2013

A118022 Triangle T, read by rows, such that T^2 = SHIFT-UP(T); i.e., the matrix square of T shifts each column of T up 1 row, dropping the main diagonal consisting of the powers of 2: [T^2](n,k) = T(n+1,k) with T(n,n) = 2^n for n>=k>=0.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 19, 24, 16, 8, 243, 304, 192, 64, 16, 6227, 7776, 4864, 1536, 256, 32, 319251, 398528, 248832, 77824, 12288, 1024, 64, 32737427, 40864128, 25505792, 7962624, 1245184, 98304, 4096, 128, 6714170259, 8380781312, 5230608384

Offset: 0

Paul D. Hanna, Apr 10 2006

Column 0 is A118023, where T(n,k) = A118023(n-k)*2^(k*(n-k+1)).

			Triangle T begins:
1;
1,2;
3,4,4;
19,24,16,8;
243,304,192,64,16;
6227,7776,4864,1536,256,32;
319251,398528,248832,77824,12288,1024,64;
32737427,40864128,25505792,7962624,1245184,98304,4096,128; ...
Matrix square, T^2, equals SHIFT_UP(T):
1;
3,4;
19,24,16;
243,304,192,64;
6227,7776,4864,1536,256;
319251,398528,248832,77824,12288,1024; ...
G.f. for column 0: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-4x) + ...
+ T(n,0)*x^n*(1-x)(1-2x)(1-4x)*..*(1-2^n*x) + ...
G.f. for column 1: 2 = 2(1-2x) + 4*x*(1-2x)(1-4x) + 24*x^2*(1-2x)(1-4x)(1-8x) + ...
+ T(n+1,1)*x^n*(1-2x)(1-4x)(1-8x)*..*(1-2^(n+1)*x) + ...
G.f. for column 2: 4 = 4(1-4x) + 16*x*(1-4x)(1-8x) + 192*x^2*(1-4x)(1-8x)(1-16x) + ...
+ T(n+2,2)*x^n*(1-4x)(1-8x)(1-16x)*..*(1-2^(n+2)*x) + ...

Cf. A118023 (column 0); A117401 (related triangle); A118024 (variant).

{T(n, k)=local(A=matrix(1, 1), B); A[1, 1]=1; for(m=2, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, i]=2^(i-1), if(j==1, B[i, j]=(A^2)[i-1, 1], B[i, j]=(A^2)[i-1, j])); )); A=B); return(A[n+1, k+1])}

G.f. for column k: 2^k = Sum{n>=0} T(n+k,k)*x^n*prod_{j=0..n} (1-2^(j+k)*x). T(n,k) = T(n-k,0)*2^(k*(n-k+1)) = A118023(n-k)*2^(k*(n-k+1)).

A118024 Triangle T, read by rows, T(n,k) = T(n-k)2^(k(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)2^(k(n-k)) for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 8, 4, 1, 28, 48, 32, 8, 1, 216, 448, 384, 128, 16, 1, 2864, 6912, 7168, 3072, 512, 32, 1, 66656, 183296, 221184, 114688, 24576, 2048, 64, 1, 2760896, 8531968, 11730944, 7077888, 1835008, 196608, 8192, 128, 1, 205824384, 706789376

Offset: 0

Paul D. Hanna, Apr 10 2006

Column 0 is A118025, where T(n,k) = A118025(n-k)*2^(k*(n-k)).

			Triangle T begins:
1;
1,1;
2,2,1;
6,8,4,1;
28,48,32,8,1;
216,448,384,128,16,1;
2864,6912,7168,3072,512,32,1;
66656,183296,221184,114688,24576,2048,64,1; ...
2760896,8531968,11730944,7077888,1835008,196608,8192,128,1; ...
Matrix square is given by [T^2](n,k) = T(n-k+1,0)*2^(k*(n-k)):
1;
2,1;
6,4,1;
28,24,8,1;
216,224,96,16,1;
2864,3456,1792,384,32,1; ...
so that column 0 of T^2 equals column 0 of T shift left 1 place.

Cf. A118025 (column 0); A117401 (related triangle); A118022 (variant).

Cf. A123305.

{T(n, k)=if(n<0 || k>n,0,if(n==k,1,2^k*sum(j=0, n-1, T(n-1, j)*T(j, k)); ))} \\ Paul D. Hanna, Sep 25 2006

T(n,k) = A118025(n-k)*2^(k*(n-k)) for n>=k>=0.

A134530 Matrix log of triangle A111636, where A111636(n,k) = (2^k)^(n-k)*C(n,k) for n>=k>=0.

Original entry on oeis.org

0, 1, 0, -1, 4, 0, 5, -12, 12, 0, -79, 160, -96, 32, 0, 3377, -6320, 3200, -640, 80, 0, -362431, 648384, -303360, 51200, -3840, 192, 0, 93473345, -162369088, 72619008, -11325440, 716800, -21504, 448, 0, -56272471039, 95716705280, -41566486528, 6196822016, -362414080, 9175040, -114688, 1024, 0

Offset: 0

Paul D. Hanna, Oct 30 2007

			Triangle begins:
0,
1, 0;
-1, 4, 0;
5, -12, 12, 0;
-79, 160, -96, 32, 0;
3377, -6320, 3200, -640, 80, 0;
-362431, 648384, -303360, 51200, -3840, 192, 0;
93473345, -162369088, 72619008, -11325440, 716800, -21504, 448, 0; ...
Matrix exponentiation yields triangle A111636, which begins:
1;
1, 1;
1, 4, 1;
1, 12, 12, 1;
1, 32, 96, 32, 1;
1, 80, 640, 640, 80, 1; ...

Cf. A134531 (column 0); related triangles: A111636, A117401; A011266.

{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,2^((c-1)*(r-c))*binomial(r-1,c-1))),L); L=sum(i=1,#M,-(M^0-M)^i/i);L[n+1,k+1]}

T(n,k) = A134531(n-k)*(2^k)^(n-k)*C(n,k), where A134531 is column 0 and satisfies: G.f.: Sum_{n>=0} A134531(n)*x^n/[n!*2^(n*(n-1)/2)] = log(Sum_{n>=0}x^n/[n!*2^(n*(n-1)/2)]).

A118410 G.f. A(x) = Sum_{n>=0} a(n)x^n/2^(n(n-1)/2) satisfies: A(x) = Sum_{n>=0} A(x)^nx^n/2^(n(n-1)/2).

Original entry on oeis.org

1, 1, 3, 21, 321, 10385, 699073, 96908737, 27478721537, 15863659383041, 18583701166494721, 44066148876930001921, 211105432749968736673793, 2040201553888722742048509953, 39729701298130761785818052935681

Offset: 0

Paul D. Hanna, Apr 27 2006

nonn

			A(x) = 1 + x + 3*x^2/2 + 21*x^3/8 + 321*x^4/64 + 10385*x^5/1024 +...
A(x) = 1 + x*A(x) + x^2*A(x)^2/2 + x^3*A(x)^3/8 +...

Cf. A117401.

{a(n)=2^(n*(n-1)/2)*polcoeff(1/x*serreverse(x/sum(k=0,n,x^k/2^(k*(k-1)/2)+x*O(x^n))),n)}

G.f.: A(x) = (1/x)*series_reversion[x/Sum_{n>=0} x^n/2^(n*(n-1)/2)].

A344260 a(n) is the number of relations from an n-element set into a set of at most n elements.

Original entry on oeis.org

1, 3, 21, 585, 69905, 34636833, 69810262081, 567382630219905, 18519084246547628289, 2422583247133816584929793, 1268889750375080065623288448001, 2659754699919401766201267083003561985, 22306191045953951743035482794815064402563073, 748380193317489370459454048174977015562807531282433

Offset: 0

Stefano Spezia, May 13 2021

nonn

Symmetrically, also the number of relations from a set of at most n elements into an n-element set.

Row sums of A344110.

Cf. A000079, A002416, A031971, A089072, A117401, A275779.

Join[{1},Table[(2^(n+n^2)-1)/(2^n-1),{n,13}]]

a(n) = (2^(n+n^2) - 1)/(2^n - 1) for n > 0 and a(0) = 1.
a(n) ~ 2^(n^2).
a(n) = A275779(n) + 1. - Hugo Pfoertner, May 14 2021

A368264 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder by two distinct tiles.

Original entry on oeis.org

2, 4, 3, 8, 10, 4, 16, 36, 24, 6, 32, 136, 176, 70, 8, 64, 528, 1376, 1044, 208, 14, 128, 2080, 10944, 16456, 6560, 700, 20, 256, 8256, 87424, 262416, 209728, 43800, 2344, 36, 512, 32896, 699136, 4195360, 6710912, 2796976, 299600, 8230, 60

Offset: 1

Peter Kagey, Dec 21 2023

			Table begins:
  n\k|  1   2     3       4         5           6
  ---+-------------------------------------------
   1 |  2   4     8      16        32          64
   2 |  3  10    36     136       528        2080
   3 |  4  24   176    1376     10944       87424
   4 |  6  70  1044   16456    262416     4195360
   5 |  8 208  6560  209728   6710912   214748416
   6 | 14 700 43800 2796976 178962784 11453291200

Peter Kagey, Illustration of T(2,3)=36
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A117401, A368257, A368259, A368261, A368263.

A368264[n_, m_] := 1/n (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &])

cp's OEIS Frontend

A176644 Triangle T(n, k) = 40^(k*(n-k)), read by rows.

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A117403 a(n) = Sum_{k=0..floor(n/2)} 2^((n-2*k)*k) for n>=0.

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A118022 Triangle T, read by rows, such that T^2 = SHIFT-UP(T); i.e., the matrix square of T shifts each column of T up 1 row, dropping the main diagonal consisting of the powers of 2: [T^2](n,k) = T(n+1,k) with T(n,n) = 2^n for n>=k>=0.

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A118024 Triangle T, read by rows, T(n,k) = T(n-k)*2^(k*(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)*2^(k*(n-k)) for n>=k>=0.

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A134530 Matrix log of triangle A111636, where A111636(n,k) = (2^k)^(n-k)*C(n,k) for n>=k>=0.

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A118410 G.f. A(x) = Sum_{n>=0} a(n)*x^n/2^(n*(n-1)/2) satisfies: A(x) = Sum_{n>=0} A(x)^n*x^n/2^(n*(n-1)/2).

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A344260 a(n) is the number of relations from an n-element set into a set of at most n elements.

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Mathematica

Formula

A368264 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder by two distinct tiles.

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A117403 a(n) = Sum_{k=0..floor(n/2)} 2^((n-2k)k) for n>=0.

A118024 Triangle T, read by rows, T(n,k) = T(n-k)2^(k(n-k)) such that column 0 of the matrix square of T equals column 0 of T shifted left: [T^2](n,k) = T(n-k+1,0)2^(k(n-k)) for n>=k>=0.

A118410 G.f. A(x) = Sum_{n>=0} a(n)x^n/2^(n(n-1)/2) satisfies: A(x) = Sum_{n>=0} A(x)^nx^n/2^(n(n-1)/2).