A011379 -id:A011379 - cp's OEIS frontend

A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

1, 0, 1, 0, 2, 2, 0, 3, 12, 6, 0, 4, 36, 72, 24, 0, 5, 80, 360, 480, 120, 0, 6, 150, 1200, 3600, 3600, 720, 0, 7, 252, 3150, 16800, 37800, 30240, 5040, 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320, 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880

Offset: 0

Peter Luschny, Mar 28 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,   2;
  [3] 0, 3,  12,     6;
  [4] 0, 4,  36,    72,     24;
  [5] 0, 5,  80,   360,    480,     120;
  [6] 0, 6, 150,  1200,   3600,    3600,     720;
  [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;
  [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;
  [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;

Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.

Cf. A144084, A021010.

A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
seq(seq(A361893(n,k), k = 0..n), n = 0..9);
# Using the egf.:
egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);

T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8k))/2) and with 0^0 = 1.

Original entry on oeis.org

2, 1, 2, 0, 2, 2, 1, 2, 3, 2, 0, 2, 4, 4, 2, 0, 2, 5, 6, 5, 2, 1, 2, 6, 10, 8, 6, 2, 0, 2, 8, 12, 17, 10, 7, 2, 0, 2, 9, 18, 20, 26, 12, 8, 2, 0, 2, 10, 28, 32, 30, 37, 14, 9, 2, 1, 2, 12, 30, 65, 50, 42, 50, 16, 10, 2, 0, 2, 16, 36, 68, 126, 72, 56, 65, 18, 11, 2, 0, 2, 17, 54, 80, 130

Offset: 0

Henry Bottomley, Jun 22 2000

			a(50) = T(5,4) = 5^2+5^1 = 30

Rows include A010054 (apart from initial term), A007395 and A048645 (offset). Subsequent rows are sums of two powers of a given number and also rewritings of A052216 from a particular base to base 10. Columns include A007395, A000027, A005843, A002522, A002378, A001105, A001093, A034262, A011379, A033431, A002523.

T(n, k) = n^A025581(k)+n^A002262(k)

A119580 a(n) = (n^2+n^3)binomial(2n,n).

Original entry on oeis.org

0, 4, 72, 720, 5600, 37800, 232848, 1345344, 7413120, 39382200, 203231600, 1024287264, 5062180032, 24607819600, 117942804000, 558423072000, 2615901857280, 12139419556440, 55866532906800, 255192804636000, 1157910842088000, 5222177897816880, 23422829664131040

Offset: 0

Zerinvary Lajos, May 31 2006

Cf. A000108, A000984, A001622, A011379.

[seq ((n^2+n^3)*(binomial(2*n,n)),n=0..29)];

Table[(n^2 + n^3) * Binomial[2 n, n], {n, 0, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)

From Amiram Eldar, Aug 28 2022: (Start)
a(n) = (n*(n+1))^2*A000108(n).
Sum_{n>=1} 1/a(n) = Pi/sqrt(3) - Pi^2/18 - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(phi)^2 - 2*sqrt(5)*log(phi) + 1, where phi is the golden ratio (A001622). (End)
a(n) = A000984(n)*A011379(n). - Michel Marcus, Aug 28 2022

A119582 a(n) = (n^2+n^3)(binomial(2n,n))/2.

Original entry on oeis.org

0, 2, 36, 360, 2800, 18900, 116424, 672672, 3706560, 19691100, 101615800, 512143632, 2531090016, 12303909800, 58971402000, 279211536000, 1307950928640, 6069709778220, 27933266453400, 127596402318000, 578955421044000, 2611088948908440, 11711414832065520, 52265818258804800

Offset: 0

Zerinvary Lajos, May 31 2006

Cf. A000984, A001622, A011379.

[seq ((n^2+n^3)*(binomial(2*n,n))/2,n=0..29)];

a[n_] := (n^2 + n^3) * Binomial[2*n, n]/2; Array[a, 30, 0] (* Amiram Eldar, Sep 04 2022 *)

From Amiram Eldar, Sep 04 2022: (Start)
a(n) = A011379(n)*A000984(n)/2.
Sum_{n>=1} 1/a(n) = 2*Pi/sqrt(3) - Pi^2/9 - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - 4*sqrt(5)*log(phi) + 12*log(phi)^2, where phi is the golden ratio (A001622). (End)

A154733 Integers of the form k = m^3+m^2 such that k-+1 are primes.

Original entry on oeis.org

12, 150, 810, 1452, 1872, 18252, 87120, 106032, 242172, 333270, 362952, 712890, 1968750, 2763600, 3198132, 5767380, 10551420, 16451580, 18469440, 23230350, 25498620, 30468672, 31955502, 34752252, 39768102, 40825920, 56770560

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 14 2009

nonn

			2^3+2^2 = 12, and 12-+1 are primes, so 12 is a term.

Cf. A111503, A154731, A154732

lst={};Do[k=n^3+n^2;If[PrimeQ[k-1]&&PrimeQ[k+1],AppendTo[lst,k]],{n,8!}];lst

A011379 INTERSECT A014574. - R. J. Mathar, Jul 16 2022

a(n) = A011379(A154732(n)). - R. J. Mathar, Jul 16 2022

A176599 Numerators of the first column of a table with top row the nonnegative integers and successive rows defined by a reverted Akiyama-Tanigawa procedure.

Original entry on oeis.org

1, -1, -1, -5, -7, -1631, -96547, -40291823, -16870575007, -7075000252463, -2969301738826267, -13713149169712887583, -10557203537780702505907

Offset: 0

Paul Curtz, Apr 21 2010

Define a reverted Akiyama-Tanigawa procedure which takes a sequence s(1), s(2), s(3), ..., as input and constructs the sequence of (s(k)-s(k+1))/k as output. (The difference from the standard algorithm is that the differences are divided by k, not multiplied by k.)
Starting from a top row with nonnegative integers, the following table is constructed row after row by applying the reverted algorithm in succession:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
-1, -1/2, -1/3, -1/4, -1/5, -1/6, -1/7, -1/8, -1/9, -1/10, -1/11, ...
-1/2, -1/12, -1/36, -1/80, -1/150, -1/252, -1/392, -1/576, -1/810, ...
-5/12, -1/36, -11/2160, -7/4800, -17/31500, -5/21168, -23/197568, ...
-7/18, -49/4320, -157/129600, -463/2016000, -803/13230000, ...
-1631/4320, -1313/259200, -17813/54432000, -35767/846720000, ...
-96547/259200, -257917/108864000, -2171917/22861440000, ...
The numerators of the left column define the current sequence.
The denominators of the third row are in A011379.

			From _Peter Bala_, Aug 14 2012: (Start)
Column 2: Sum_{n >= 2} x^(n-1)/(Product_{k = 2..n} (x-k)) = -(1/2)*x - (1/12)*x^2 - (1/36)*x^3 - (49/4320)*x^4 - ...
Column 3: Sum_{n >= 3} x^(n-2)/(Product_{k = 3..n} (x-k)) = -(1/3)*x - (1/36)*x^2 - (11/2160)*x^3 - (157/129600)*x^4 - .... (End)

Cormac O'Sullivan, Stirling's approximation and a hidden link between two of Ramanujan's approximations, arXiv:2208.02898 [math.NT], 2022. See p. 17.
Craig A. Tracy and H. Widom, On the ground state energy of the delta-function Bose gas, arXiv preprint arXiv:1601.04677 [math-ph], 2016.

Cf. A024427.

a[1, k_] := k; a[n_, k_] := a[n, k] = (a[n-1, k] - a[n-1, k+1])/k; a[n_] := Numerator[a[n, 1]]; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Aug 02 2012 *)

From Peter Bala, Aug 14 2012: (Start)
The o.g.f. for the rational numbers in the first column of the above table is Sum_{n >= 0} x^n/(Product_{k = 1..n} (x-k)) = 1 - x - 1/2*x^2 - 5/12*x^3 - 7/18*x^4 - .... This yields the formula |a(n)| = numerator of Sum_{k = 0..n-1} (1/k!) * Sum_{i = 0..k} (-1)^i*binomial(k,i)*(k-i+1)^(k-n). Cf. A024427.
More generally, the o.g.f. for the rational numbers in the r-th column of the above table (excluding the first entry of r) is Sum_{n >= r} x^(n+1-r)/ (Product_{k = r..n} (x-k)).
(End)
The first column of the above table lists the coefficients of the expansion of b(1)x/(1+b(2)x/(1+b(3)x/(1+b(4)x/(...)))), a continued fraction, where b(n) are -1, -1/2, -1/3, -1/4, ... i.e., the second row of the table above. - Benedict W. J. Irwin, May 10 2016

A178343 Triangle T(n,m)= binomial(n, m)/Beta(m + 1, n - m + 1) read by rows.

Original entry on oeis.org

1, 2, 2, 3, 12, 3, 4, 36, 36, 4, 5, 80, 180, 80, 5, 6, 150, 600, 600, 150, 6, 7, 252, 1575, 2800, 1575, 252, 7, 8, 392, 3528, 9800, 9800, 3528, 392, 8, 9, 576, 7056, 28224, 44100, 28224, 7056, 576, 9, 10, 810, 12960, 70560, 158760, 158760, 70560, 12960, 810, 10

Offset: 0

Roger L. Bagula, May 25 2010

Beta(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y) is the Beta-function.

Row sums are A037965(n+1). The second column is A011379.

			1;
2, 2;
3, 12, 3;
4, 36, 36, 4;
5, 80, 180, 80, 5;
6, 150, 600, 600, 150, 6;
7, 252, 1575, 2800, 1575, 252, 7;
8, 392, 3528, 9800, 9800, 3528, 392, 8;
9, 576, 7056, 28224, 44100, 28224, 7056, 576, 9;
10, 810, 12960, 70560, 158760, 158760, 70560, 12960, 810, 10;
11, 1100, 22275, 158400, 485100, 698544, 485100, 158400, 22275, 1100, 11;

Cf. A037965

Flatten[Table[Table[Binomial[n, m]/Beta[m + 1, n - m + 1], {m, 0, n}], {n, 0, 10}]]

T(n,m)=T(n,n-m) = (n+1)*( binomial(n,m))^2 = (n+1)*A008459(n).

Edited by the Assoc. Eds. of the OEIS - Jun 27 2010

A256560 Triangle read by rows, sums of 2 distinct nonzero squares plus sums of 2 distinct nonzero cubes: T(n,k) = n^2 + k^2 + n^3 + k^3, 1 <= k <= n-1.

Original entry on oeis.org

14, 38, 48, 82, 92, 116, 152, 162, 186, 230, 254, 264, 288, 332, 402, 394, 404, 428, 472, 542, 644, 578, 588, 612, 656, 726, 828, 968, 812, 822, 846, 890, 960, 1062, 1202, 1386, 1102, 1112, 1136, 1180, 1250, 1352, 1492, 1676, 1910

Offset: 2

Bob Selcoe, Apr 02 2015

All terms are even.
T(n,1) = A011379(n) + 2.
When n=k+1, T(n,k+1) = A011379(n-1) + A011379(n) = 2n^3 - n^2 + n.

			Triangle starts T(2,1):
n\k   1    2    3    4    5    6    7     8    9   10
2:   14
3:   38   48
4:   82   92   116
5:   152  162  186  230
6:   254  264  288  332  402
7:   394  404  428  472  542  644
8:   578  588  612  656  726  828  968
9:   812  822  846  890  960  1062 1202 1386
10:  1102 1112 1136 1180 1250 1352 1492 1676 1910
11:  1454 1464 1488 1532 1602 1704 1844 2028 2262 2552
...
The successive terms are: (2^2 + 1^2 + 2^3 + 1^3), (3^2 + 1^2 + 3^3 + 1^3), (3^2 + 2^2 + 3^3 + 2^3), (4^2 + 1^2 + 4^3 + 1^3), (4^2 + 2^2 + 4^3 + 2^3), (4^2 + 3^2 + 4^3 + 3^3), ...
T(7,4) = 472 because 7^2 + 7^3 + 4^2 + 4^3 = 472.

Cf. A055096 (sums of 2 distinct nonzero squares), A256497 (sums of 2 distinct nonzero cubes), A011379, A024670, A004431, A049450.

a(n) = A055096(n) + A256497(n-1).
T(n,k) = T055096(n,k) + T256547(n-1,k).
T(n,k) = T(n-1,k) + A049450(n).
T(n,k) = T(n,k-1) + A049450(k).
T(n,k) = A011379(n) + A011379(k).

A273788 Least number k such that k^2 + k^3 is of the form x^2 + y^3 in exactly n ways where x, y > 0.

Original entry on oeis.org

1, 6, 24, 40, 180, 440, 3640, 18480, 137280, 1320, 703560, 15960, 1256640, 1436160, 96360

Offset: 1

Altug Alkan, May 30 2016

			a(2) = 6 because 6^2 + 6^3 = 15^2 + 3^3.

Cf. A011379, A060835, A122054.

a(7)-a(15) from Giovanni Resta, Jun 03 2016

A325146 A(n, k) = Stirling2(n + k, k)A053657(n)k!/(n + k)!, array read by ascending antidiagonals for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 2, 14, 3, 1, 0, 48, 12, 30, 4, 1, 0, 16, 496, 36, 52, 5, 1, 0, 576, 288, 2064, 80, 80, 6, 1, 0, 144, 18288, 1656, 5832, 150, 114, 7, 1, 0, 3840, 8160, 145200, 5920, 13240, 252, 154, 8, 1

Offset: 0

Peter Luschny, May 22 2019

			[0] 1,   1,     1,      1,      1,       1,       1,        1, ... A000012
[1] 0,   1,     2,      3,      4,       5,       6,        7, ... A001477
[2] 0,   4,    14,     30,     52,      80,     114,      154, ... A049451
[3] 0,   2,    12,     36,     80,     150,     252,      392, ... A011379
[4] 0,  48,   496,   2064,   5832,   13240,   26088,    46536, ...
[5] 0,  16,   288,   1656,   5920,   16200,   37296,    76048, ...
[6] 0, 576, 18288, 145200, 654816, 2153280, 5775936, 13429248, ...
     A163176

Rows include A001477, A049451, A011379. Columns include A163176.

Cf. A053657.

A := (n, k) -> Stirling2(n + k, k)*A053657(n)*k!/(n + k)!:
seq(seq(A(n - k, k), k=0..n), n=0..10);

a053657[n_] := Product[p^Sum[Floor[(n-1) / ((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}];
A[n_, k_] := StirlingS2[n+k, k] a053657[n+1] k! / (n+k)!;
Table[A[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)

cp's OEIS Frontend

A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

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A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)*(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8*k))/2) and with 0^0 = 1.

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A119580 a(n) = (n^2+n^3)*binomial(2*n,n).

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A119582 a(n) = (n^2+n^3)*(binomial(2*n,n))/2.

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A154733 Integers of the form k = m^3+m^2 such that k-+1 are primes.

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A176599 Numerators of the first column of a table with top row the nonnegative integers and successive rows defined by a reverted Akiyama-Tanigawa procedure.

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A178343 Triangle T(n,m)= binomial(n, m)/Beta(m + 1, n - m + 1) read by rows.

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A256560 Triangle read by rows, sums of 2 distinct nonzero squares plus sums of 2 distinct nonzero cubes: T(n,k) = n^2 + k^2 + n^3 + k^3, 1 <= k <= n-1.

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A273788 Least number k such that k^2 + k^3 is of the form x^2 + y^3 in exactly n ways where x, y > 0.

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A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8k))/2) and with 0^0 = 1.

A119580 a(n) = (n^2+n^3)binomial(2n,n).

A119582 a(n) = (n^2+n^3)(binomial(2n,n))/2.

A325146 A(n, k) = Stirling2(n + k, k)A053657(n)k!/(n + k)!, array read by ascending antidiagonals for n >= 0 and k >= 0.