A014477 -id:A014477 - cp's OEIS frontend

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

1, 3, 5, 8, 12, 16, 20, 28, 36, 44, 48, 64, 80, 96, 112, 112, 144, 176, 208, 240, 272, 256, 320, 384, 448, 512, 576, 640, 576, 704, 832, 960, 1088, 1216, 1344, 1472, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 2816, 3328, 3840, 4352, 4864, 5376

Offset: 1

Fedor Igumnov, Jan 28 2014

1, 9, 45, 161, 497, 1409, ... is the sequence of perimeters (sum of border elements) of the triangle.
1, 5, 80, 3520, 394240, 107233280, 68629299200, ... is the sequence of determinants of the triangle.
Only the first three terms are odd.

			Triangle begins:
================================================
\k |    1     2     3     4     5     6     7
n\ |
================================================
1  |    1;
2  |    3,    5;
3  |    8,   12,   16;
4  |   20,   28,   36,   44;
5  |   48,   64,   80,   96,  112;
6  |  112,  144,  176,  208,  240,  272;
7  |  256,  320,  384,  448,  512,  576,  640;
...

Fedor Igumnov, T(n,k) for n = 1..26

Cf. A001792 (column 1), A053220 (right border). Also:
A014477, row sums;
A036826, partial sums;
A058962, central elements in odd rows;
A045623, second column;
A045891, third column;
A034007, fourth column;
A167667, subdiagonal;
A130129, second subdiagonal.

int a(int n, int k) {return (n+1)*pow(2,n-2)+(k-1)*pow(2,n-1);}

/* As triangle: */ [[(n+1)*2^(n-2)+(k-1)*2^(n-1): k in [1..n]]: n in [1..10]]; // Bruno Berselli, Jan 28 2014

t[n_, k_] := (n + 1)*2^(n - 2) + (k - 1)*2^(n - 1); Table[t[n, k], {n, 10}, {k, n}] // Flatten (* Bruno Berselli, Jan 28 2014 *)

T(n,k) = T(n-1,k) + T(n-1,k+1).

Sum_{k=1..n} T(n,k) = n^2*2^(n-1) = A014477(n-1).

More terms from Bruno Berselli, Jan 28 2014

A372493 G.f. A(x) satisfies A(A(x)) = Sum_{k>=1} k^2 * 2^(k-1) * x^k.

Original entry on oeis.org

0, 1, 4, 2, 12, -30, 24, 1412, -18716, 127750, -19448, -10721556, 98983992, 546122580, -19718580272, 30721006440, 4638904011364, -44204880124922, -1218793973236472, 26364933421291468, 327900715232299304, -15425392878552410820, -62258050574118828336

Offset: 0

Seiichi Manyama, May 03 2024

sign

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013.

Cf. A014477, A309509.

A095677 Triangle T(n,k), 0<=k<=n, read by rows, defined by Sum_{k = 0..n} T(n,k)x^k = Sum_{k = 0..n} binomial(n,k)(x+k)^n.

Original entry on oeis.org

1, 1, 2, 6, 8, 4, 54, 72, 36, 8, 680, 896, 480, 128, 16, 11000, 14400, 8000, 2400, 400, 32, 217392, 283392, 161280, 51840, 10080, 1152, 64, 5076400, 6598144, 3819648, 1285760, 274400, 37632, 3136, 128, 136761984, 177373184, 103993344

Offset: 0

Philippe Deléham Jul 04 2004

			1; 1, 2; 6, 8, 4; 54, 72, 36, 8; 680, 896, 480, 128, 16; ...

T(n, k) = binomial(n, k)*Sum_{j = 0..n} = binomial(n, j)*j^(n-k).
T(n, n) = 2^n, see A000079.
T(n+1, n) = (n+1)^2*2^n, see A014477.
T(n, 0) = n*Sum_{k = 0..n-1} T(n-1, k).

A176676 Primes not of the form prime + k^2*2^(k-1).

Original entry on oeis.org

2, 5, 7, 17, 23, 29, 71, 101, 113, 127, 179, 191, 197, 223, 239, 251, 257, 281, 283, 311, 331, 337, 359, 401, 449, 463, 491, 521, 541, 569, 587, 617, 647, 659, 673, 701, 733, 739, 743, 757, 761, 827, 853, 877, 881, 883, 907, 911, 929, 953, 997, 1051, 1063, 1093

Offset: 1

Juri-Stepan Gerasimov, Apr 23 2010

nonn

Cf. A014477.

From R. J. Mathar, Apr 27 2010: (Start)
A014477 := proc(n) (n+1)^2*2^n ; end proc:
isA176676 := proc(p) local n,a77 ; if not isprime(p) then return false; end if; for n from 1 do a77 := A014477(n-1) ; if isprime(p-a77) then return false; elif p-a77<2 then return true; end if; end do: end proc:
for n from 1 to 300 do p := ithprime(n) ; if isA176676(p) then printf("%d,",p) ; end if; end do: (End)

Corrected (149 replaced by 179, 269 removed, 643 removed, 743 and 911 and 1053 inserted) by R. J. Mathar, Apr 27 2010

cp's OEIS Frontend

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

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A372493 G.f. A(x) satisfies A(A(x)) = Sum_{k>=1} k^2 * 2^(k-1) * x^k.

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A095677 Triangle T(n,k), 0<=k<=n, read by rows, defined by Sum_{k = 0..n} T(n,k)x^k = Sum_{k = 0..n} binomial(n,k)(x+k)^n.

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A176676 Primes not of the form prime + k^2*2^(k-1).

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cp's OEIS Frontend

A236538 Triangle read by rows: T(n,k) = (n+1)*2^(n-2)+(k-1)*2^(n-1) for 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Magma

Mathematica

Formula

Extensions

A372493 G.f. A(x) satisfies A(A(x)) = Sum_{k>=1} k^2 * 2^(k-1) * x^k.

Views

Author

Keywords

Links

Crossrefs

A095677 Triangle T(n,k), 0<=k<=n, read by rows, defined by Sum_{k = 0..n} T(n,k)*x^k = Sum_{k = 0..n} binomial(n,k)*(x+k)^n.

Views

Author

Keywords

Examples

Formula

A176676 Primes not of the form prime + k^2*2^(k-1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Extensions

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

A095677 Triangle T(n,k), 0<=k<=n, read by rows, defined by Sum_{k = 0..n} T(n,k)x^k = Sum_{k = 0..n} binomial(n,k)(x+k)^n.