A002417 -id:A002417 - cp's OEIS frontend

A071238 a(n) = n(n+1)(2*n^2+1)/6.

0, 1, 9, 38, 110, 255, 511, 924, 1548, 2445, 3685, 5346, 7514, 10283, 13755, 18040, 23256, 29529, 36993, 45790, 56070, 67991, 81719, 97428, 115300, 135525, 158301, 183834, 212338, 244035, 279155, 317936, 360624, 407473, 458745, 514710, 575646, 641839

Offset: 0

N. J. A. Sloane, Jun 12 2002

Binomial transform of [1, 8, 21, 22, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
For n > 0, a(n) is the n-th antidiagonal sum of the convolution arrays A213752 and A213836). - Clark Kimberling, Jun 20 2012
The first differences are given in A277229, as a convolution of the odd-indexed triangular numbers A000217(2*n+1) and the squares A000290(n), n >= 0. - J. M. Bergot, Sep 14 2016

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000292, A002417, A071270, A277229 (first differences).

[n*(n+1)*(2*n^2+1)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

A071238:=n->n*(n+1)*(2*n^2+1)/6: seq(A071238(n), n=0..60); # Wesley Ivan Hurt, Sep 24 2016

Table[n (n + 1) (2 n^2 + 1)/6, {n, 0, 37}] (* or *)
CoefficientList[Series[x (1 + x) (1 + 3 x)/(1 - x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Sep 14 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,9,38,110},40] (* Harvey P. Dale, Oct 02 2021 *)

a(n)=n*(n+1)*(2*n^2+1)/6; \\ Joerg Arndt, Sep 04 2013

G.f.: x*(1+x)*(1+3*x)/(1-x)^5. - Colin Barker, Mar 22 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4, a(0)=0, a(1)=1, a(2)=9, a(3)=38, a(4)=110. - Yosu Yurramendi, Sep 03 2013
E.g.f.: (1/6)*x*(6 + 21*x + 14*x^2 + 2*x^3)*exp(x). - G. C. Greubel, Sep 17 2016
a(n) = n*A000292(n) + (n-1)*A000292(n-1). - Bruno Berselli, Sep 22 2016
a(n) = A002417(n-1) + A002417(n). - Yasser Arath Chavez Reyes, Feb 15 2024

A076389 Sum of squares of numbers that cannot be written as tn + u(n+1) for nonnegative integers t,u.

Original entry on oeis.org

0, 1, 30, 220, 950, 3045, 8036, 18480, 38340, 73425, 131890, 224796, 366730, 576485, 877800, 1300160, 1879656, 2659905, 3693030, 5040700, 6775230, 8980741, 11754380, 15207600, 19467500, 24678225, 31002426, 38622780, 47743570

Offset: 1

Floor van Lamoen, Oct 09 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002417, A076460-A076465.

Maple
```
seq(n^2*(n^2-1)*(n^2-n-1)/12,n=1..40);
```

n^2*(n^2-1)*(n^2-n-1)/12.

G.f.:(1+23*x+31*x^2+5*x^3)*x^2/(1-x)^7

A076459 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly n ways.

Original entry on oeis.org

1, 57, 390, 1510, 4335, 10311, 21532, 40860, 72045, 119845, 190146, 290082, 428155, 614355, 860280, 1179256, 1586457, 2099025, 2736190, 3519390, 4472391, 5621407, 6995220, 8625300, 10545925, 12794301, 15410682, 18438490, 21924435, 25918635, 30474736, 35650032

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

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

Cf. A002417, A076454-A076458.

[n*(n+1)*(2*n^3+2*n^2-2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/2*n*(n+1)*(2*n^3+2*n^2-2*n-1),n=1..35);

CoefficientList[Series[(1 + 51 x + 63 x^2 + 5 x^3)/(1 - x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(2*n^3+2*n^2-2*n-1)/2.

G.f.: x*(1+51*x+63*x^2+5*x^3)/(1-x)^6.

More terms from Vincenzo Librandi, Dec 30 2013

A060102 Bisection of triangle A060098: even-indexed members of column sequences of A060098 (not counting leading zeros).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 8, 1, 1, 16, 30, 13, 1, 1, 25, 80, 71, 19, 1, 1, 36, 175, 259, 140, 26, 1, 1, 49, 336, 742, 660, 246, 34, 1, 1, 64, 588, 1806, 2370, 1442, 399, 43, 1, 1, 81, 960, 3906, 7062, 6292, 2828, 610, 53

Offset: 0

Wolfdieter Lang, Apr 06 2001

Row sums give A052975. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A000290 (squares), A002417(n+1), A060103-5.

Companion triangle (odd-indexed members) A060556.

			{1}; {1,1}; {1,4,1}; {1,9,8,1}; ... Pe(3,x) = 1 + 3*x.

a(n, m) = A060098(2*n-m, m).
a(n, m) = Sum_{j=0..floor((m+1)/2)} binomial((n-m)-j+2*m, 2*m)*binomial(m+1, 2*j), n >= m >= 0, otherwise zero.
G.f. for column m: (x^m)*Pe(m+1, x)/(1-x)^(2*m+1), with Pe(n, x) = Sum_{j=0..floor(n/2)} binomial(n, 2*j)*x^j (even members of row n of Pascal triangle A007318).

A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 24, 12, 1, 5, 80, 90, 20, 1, 6, 240, 540, 240, 30, 1, 7, 672, 2835, 2240, 525, 42, 1, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 10, 11520, 262440, 860160, 787500, 272160, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. - Peter Luschny, Mar 13 2009
T(n,k) is the sum of the products of the cardinality of the blocks (cells) in the set partitions of {1,2,..,n} into exactly k blocks.
From Peter Bala, Jul 22 2014: (Start)
Exponential Riordan array [(1+x)*exp(x), x*exp(x)].
Let M = A093375, the exponential Riordan array [(1+x)*exp(x), x], and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
  /I_k 0\
  \ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... - see the Example section. (End)
The Bell transform of n+1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins
1;
2, 1;
3, 6, 1;
4, 24, 12, 1; ...
From _Peter Bala_, Jul 22 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1          \/1        \/1        \      /1           \
|2  1       ||0 1      ||0 1      |      |2  1        |
|3  4  1    ||0 2 1    ||0 0 1    |... = |3  6  1     |
|4  9  6 1  ||0 3 4 1  ||0 0 2 1  |      |4 24 12  1  |
|5 16 18 8 1||0 4 9 6 1||0 0 3 4 1|      |5 80 90 20 1|
|...        ||...      ||...      |      |...         | (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Idempotent Number

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248. A093375.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n+1,k+1)*(k+1)^(n-k): seq(seq(T(n, k), k=0..n), n=0..10); # Georg Fischer, Oct 27 2021

t = Transpose[ Table[ Range[0, 11]! CoefficientList[ Series[(x Exp[x])^n/n!, {x, 0, 11}], x], {n, 11}]]; Table[ t[[n, k]], {n, 2, 11}, {k, n - 1}] // Flatten (* or simply *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Table[t[n, k], {n, 10}, {k, n}] // Flatten

for(n=1, 25, for(k=1, n, print1(binomial(n,k)*k^(n-k), ", "))) \\ G. C. Greubel, Jan 05 2017

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: n+1, 10) # Peter Luschny, Jan 18 2016

A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.

Original entry on oeis.org

1, 6, 1, 15, 7, 1, 28, 22, 8, 1, 45, 50, 30, 9, 1, 66, 95, 80, 39, 10, 1, 91, 161, 175, 119, 49, 11, 1, 120, 252, 336, 294, 168, 60, 12, 1, 153, 372, 588, 630, 462, 228, 72, 13, 1, 190, 525, 960, 1218, 1092, 690, 300, 85, 14, 1, 231, 715, 1485, 2178, 2310, 1782, 990, 385, 99, 15, 1

Offset: 0

Gary W. Adamson, Nov 24 2006

Left border = A000384, hexagonal numbers. The following columns are A002412, A002417, A034263, A051947, ...
Row sums = (1, 7, 23, 59, 135, 291, ...) = A126284.
A125232 is the analogous triangle for the pentagonal numbers.

			First few rows of the triangle:
   1;
   6,   1;
  15,   7,   1;
  28,  22,   8,   1;
  45,  50,  30,   9,  1;
  66,  95,  80,  39, 10,  1;
  91, 161, 175, 119, 49, 11, 1;
  ...
Example: (5,3) = 80 = 30 + 50 = (4,3) + (4,2).

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1964, p. 189.

Cf. A000384, A002412, A002417, A034263, A051947, A125232.

A000384Psum:= proc(n,k) coeftayl( x*(1+3*x)/(1-x)^(3+k),x=0,n) ; end: A125233 := proc(n,k) A000384Psum(n-k,k) ; end: for n from 1 to 15 do for k from 0 to n -1 do printf("%d,",A125233(n,k)) ; od: od: # R. J. Mathar, May 03 2008

T[n_, k_] := T[n, k] = Which[k == 0, n (2 n - 1), 1 <= k < n, T[n - 1, k] + T[n - 1, k - 1], True, 0];
Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Sep 14 2023, after R. J. Mathar *)

T(n,0)=A000384(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>1. - R. J. Mathar, May 03 2008

Edited and extended by R. J. Mathar, May 03 2008, and M. F. Hasler, Sep 29 2012

A213750 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 5, 3, 14, 11, 5, 30, 26, 17, 7, 55, 50, 38, 23, 9, 91, 85, 70, 50, 29, 11, 140, 133, 115, 90, 62, 35, 13, 204, 196, 175, 145, 110, 74, 41, 15, 285, 276, 252, 217, 175, 130, 86, 47, 17, 385, 375, 348, 308, 259, 205, 150, 98, 53, 19, 506, 495, 465, 420

Offset: 1

Clark Kimberling, Jun 20 2012

Principal diagonal:  A007585
Antidiagonal sums:  A002417
row 1,  (1,2,3,4,5,...)**(1,3,5,7,9,...): A000330
row 2,  (1,2,3,4,5,...)**(3,5,7,9,...): A051925
row 3,  (1,2,3,4,5,...)**(5,7,9,11,...): (2*k^3 + 15*k^2 + 13*k)/6
row 4,  (1,2,3,4,5,...)**(7,9,11,13,...): (2*k^3 + 21*k^2 + 19*k)/6
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....14...30....55....91
3....11...26...50....85....133
5....17...38...70....115...175
7....23...50...90....145...217
9....29...62...110...175...259
11...35...74...130...205...301

Cf. A213500.

b[n_] := n; c[n_] := 2 n - 1;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213750 *)
d = Table[t[n, n], {n, 1, 40}] (* A007585 *)
s1 = Table[s[n], {n, 1, 50}] (* A002417 *)
FindLinearRecurrence[s1]
FindGeneratingFunction[s1, x]

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = (2*n - 1) - (2*n - 3)*x and g(x) = (1 - x )^4.

A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 24, 4, 0, 1, 20, 90, 80, 5, 0, 1, 30, 240, 540, 240, 6, 0, 1, 42, 525, 2240, 2835, 672, 7, 0, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 0, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 0, 1, 90, 2880, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1,
1,  0,
1,  2,   0,
1,  6,   3,    0,
1, 12,  24,    4,    0,
1, 20,  90,   80,    5,   0,
1, 30, 240,  540,  240,   6, 0,
1, 42, 525, 2240, 2835, 672, 7, 0,
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

There are 4 versions: A059297-A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.

/* As triangle: */ [[Binomial(n,k)*(n-k)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

T := (n, k) -> binomial(n, k) * (n - k)^k:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

t[n_, k_] := Binomial[n, k]*(n - k)^k; Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

concat([1], for(n=0, 25, for(k=0, n, print1(binomial(n,k)*(n-k)^k, ", ")))) \\ G. C. Greubel, Jan 05 2017

Name corrected by Peter Luschny, Nov 12 2023

A068253 1/3 of the number of colorings of an n X n square array with 3 colors.

Original entry on oeis.org

1, 6, 82, 2604, 193662, 33865632, 13956665236, 13574876544396, 31191658416342674, 169426507164530254380, 2176592549084872196370724, 66158464020552857153017287240, 4759146677426447759184119036493676, 810410082813497381147177065840601910384

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..24

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.
See A047938 for number of improper colorings.
Main diagonal of A078099.
Twice A207993 for n>1.

M[1] = {{1}}; M[m_] := M[m] = {{M[m - 1], Transpose[M[m - 1]]}, {Array[0 &, {2^(m - 2), 2^(m - 2)}], M[m - 1]}} // ArrayFlatten; W[m_] := M[m] + Transpose[M[m]]; T[m_, 1] := 2^(m - 1); T[1, n_] := 2^(n - 1); T[m_, n_] := MatrixPower[W[m], n - 1] // Flatten // Total; a[n_] := T[n, n]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 01 2017, after code from A078099 *)

For formula see A078099.

More terms from Vladeta Jovovic, Jul 22 2004
a(11)-a(12) from Alois P. Heinz, Mar 25 2009
a(13)-a(14) from Andrew Howroyd, Jun 26 2017

A068271 1/4 the number of colorings of an n X n rhombic hexagonal array with 4 colors.

Original entry on oeis.org

1, 12, 264, 11424, 1008576, 184910592, 71033971200, 57469424744448, 98237339264864256, 355574469749489123328, 2729407814499050197254144, 44482040254775494064841818112, 1540473331004371306422199656382464, 113440401780206156918876627438624833536

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Terms for rhombic- and staggered- hexagonal arrays are the same for n in 1..4.

Andrew Howroyd, Table of n, a(n) for n = 1..16

Cf. A068239-A068305, A000332, A002417, A027441, A212162, A212163.

a(9) from Alois P. Heinz, May 02 2012

a(10)-a(14) from Andrew Howroyd, Jun 25 2017

cp's OEIS Frontend

A071238 a(n) = n*(n+1)*(2*n^2+1)/6.

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A076389 Sum of squares of numbers that cannot be written as t*n + u*(n+1) for nonnegative integers t,u.

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A076459 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly n ways.

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A060102 Bisection of triangle A060098: even-indexed members of column sequences of A060098 (not counting leading zeros).

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A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

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A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.

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A213750 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution.

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A071238 a(n) = n(n+1)(2*n^2+1)/6.

A076389 Sum of squares of numbers that cannot be written as tn + u(n+1) for nonnegative integers t,u.

A076459 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly n ways.