A030662 -id:A030662 - cp's OEIS frontend

A308292 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 16, 19, 4, 1, 1, 65, 271, 69, 5, 1, 1, 326, 7365, 5248, 251, 6, 1, 1, 1957, 326011, 1107697, 110251, 923, 7, 1, 1, 13700, 21295783, 492911196, 191448941, 2435200, 3431, 8, 1, 1, 109601, 1924223799, 396643610629, 904434761801, 35899051101, 55621567, 12869, 9, 1

Offset: 0

Seiichi Manyama, May 19 2019

For r > 1, row r is asymptotic to sqrt(2*Pi) * (r*n)^(r*n + 1/2) / ((r!)^n * exp(r*n-1)). - Vaclav Kotesovec, May 24 2020

			For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + 2*x + 4*x^2/2 + 8*x^3/6 + 14*x^4/24 + 20*x^5/120 + 20*x^6/720. So A(3,2) = 1 + 2 + 4 + 8 + 14 + 20 + 20 = 69.
Square array begins:
   1, 1,    1,        1,             1,                   1, ...
   1, 2,    5,       16,            65,                 326, ...
   1, 3,   19,      271,          7365,              326011, ...
   1, 4,   69,     5248,       1107697,           492911196, ...
   1, 5,  251,   110251,     191448941,        904434761801, ...
   1, 6,  923,  2435200,   35899051101,    1856296498826906, ...
   1, 7, 3431, 55621567, 7101534312685, 4098746255797339511, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Columns k=0..4 give A000012, A000027(n+1), A030662(n+1), A144660, A144661.
Rows n=0..4 give A000012, A000522, A003011, A308294, A308295.
Main diagonal gives A274762.
Cf. A144510.

A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * x^i/i! = (Sum_{i=0..n} x^i/i!)^k.

A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!j!k!*l!).

Original entry on oeis.org

1, 65, 7365, 1107697, 191448941, 35899051101, 7101534312685, 1458965717496881, 308290573348183629, 66577182435768923245, 14629025943480502591445, 3260160391173522631759533, 735119604833362632050789701, 167408468505328518543519208949, 38448088693846486556578015883325

Offset: 0

N. J. A. Sloane, Feb 01 2009

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..50 from Seiichi Manyama)
Vaclav Kotesovec, Recurrence (of order 4)
Vidunas, Raimundas Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).

Cf. A030662, A144660, A144662, A307324.

f:=n->add( add( add( add( (i+j+k+l)!/(i!*j!*k!*l!), i=0..n),j=0..n),k=0..n),l=0..n); [seq(f(n),n=0..20)];

Table[Sum[(i + j + k + l)! / (i!*j!*k!*l!), {i, 0, n}, {j, 0, n}, {k, 0, n}, {l, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)
Table[Sum[(1 + j + k + l + n)!/((1 + j + k + l)*j!*k!*l!), {j, 0, n}, {k, 0, n}, {l, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2019 *)
Table[Sum[(1 + k + l + 2*n)! * HypergeometricPFQ[{1, -1 - k - l - n, -n}, {-1 - k - l - 2*n, -k - l - n}, 1] / ((1 + k + l + n)*k!*l!*n!), {k, 0, n}, {l, 0, n}]/n!, {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2019 *)

{a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, sum(l=0, n, (i+j+k+l)!/(i!*j!*k!*l!)))))} \\ Seiichi Manyama, Apr 02 2019

a(n) ~ 2^(8*n + 15/2) / (81 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 02 2019

A181991 n-alternating permutations of length 4n.

Original entry on oeis.org

1, 1385, 315523, 60376809, 11593285251, 2301250545971, 472105349529479, 99537885358650089, 21451428576293883859, 4705284467293276073635, 1047067375984978044542143, 235809039854522043890582835, 53644722291938408687646120103, 12309355014854205055828909176039

Offset: 1

Peter Luschny, Apr 05 2012

nonn

a(n) = A181985(n,4).

Peter Luschny, An old operation on sequences: the Seidel transform.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Cf. A181985, A030662, A211213.

A181991 := proc(n) local E, dim, i, k; dim := 4*n;
E := array(0..dim, 0..dim); E[0, 0] := 1;
for i from 1 to dim do
if i mod n = 0 then E[i, 0] := 0 ;
   for k from i-1 by -1 to 0 do E[k, i-k] := E[k+1, i-k-1] + E[k, i-k-1] od;
else E[0, i] := 0;
   for k from 1 by 1 to i do E[k, i-k] := E[k-1, i-k+1] + E[k-1, i-k] od;
fi od; E[0, dim] end:
seq(A181991(n), n = 1..14);
# Alternatively:
a := (x) -> (4*x)!*(-1/(4*x)!+2/x!/(3*x)!+1/(2*x)!^2-3/x!^2/(2*x)!+1/x!^4):
seq(a(n), n=1..14); # Peter Luschny, Aug 13 2015

A181985[n_, len_] := Module[{e, dim = n*(len - 1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0, e[i, 0] = 0; For[k = i - 1, k >= 0, k--, e[k, i - k] = e[k + 1, i - k - 1] + e[k, i - k - 1]], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i - k] = e[k - 1, i - k + 1] + e[k - 1, i - k]]]]; Table[e[0, n*k], {k, 0, len - 1}]]; a[n_] := A181985[n, 4 + 1][[4 + 1]]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 17 2013, after Maple code in A181985 *)

a(n) = (4*n)!*(-1/(4*n)! + 2/(n!*(3*n)!) + 1/(2*n)!^2 - 3/(n!^2*(2*n)!) + 1/n!^4). - Peter Luschny, Aug 13 2015

A206735 Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 7, 21, 35, 35, 21, 7, 1, 0, 8, 28, 56, 70, 56, 28, 8, 1, 0, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Philippe Deléham, Feb 11 2012

A103452*A007318 as infinite lower triangular matrices.

Essentially the same as A199011.

			Triangle begins :
1
0, 1
0, 2, 1
0, 3, 3, 1
0, 4, 6, 4, 1
0, 5, 10, 10, 5, 1
0, 6, 15, 20, 15, 6, 1
0, 7, 21, 35, 35, 21, 7, 1
0, 8, 28, 56, 70, 56, 28, 8, 1
0, 9, 36, 84, 126, 126, 84, 36, 9, 1
0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1
0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Cf. A007318, A000071 (antidiagonal sums).

T(n,k) = A007318(n,k) - A073424(n,k).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (1+x)^n - 1 + 0^n.
T(n,0) = 0^n = A000007(n), T(n,k) = binomial(n,k) for k>0.
G.f.: (1-2*x+(1+y)*x^2)/(1-2x+(1+y)*x^2-y*x).
Sum{k, 0<=k<=n} T(n,k)^x = A000027(n+1), A000225(n), A030662(n), A096191(n), A096192(n) for x = 0, 1, 2, 3, 4 respectively.

A211213 n-alternating permutations of length 3n.

Original entry on oeis.org

1, 61, 1513, 33661, 750751, 17116009, 398840401, 9464040829, 227864057851, 5550936701311, 136526608389601, 3384729259165801, 84478081828015513, 2120572560190269841, 53494979095639780513, 1355345459896317255037, 34469858667289041256051, 879619727291950363099291

Offset: 1

Peter Luschny, Apr 05 2012

nonn

a(n) = A181985(n,3).

Peter Luschny, An old operation on sequences: the Seidel transform

Cf. A181985, A030662, A181991.

A211213 := proc(n) local E, dim, i, k; dim := 3*n;
E := array(0..dim, 0..dim); E[0, 0] := 1;
for i from 1 to dim do
if i mod n = 0 then E[i, 0] := 0 ;
   for k from i-1 by -1 to 0 do E[k, i-k] := E[k+1, i-k-1] + E[k, i-k-1] od;
else E[0, i] := 0;
   for k from 1 by 1 to i do E[k, i-k] := E[k-1, i-k+1] + E[k-1, i-k] od;
fi od; E[0, dim] end:
seq(A211213(n), n = 1..18);
# Alternatively:
a := x -> (3*x)!*(1/(3*x)!-2/(x!*(2*x)!)+1/(x!)^3):
seq(a(n),n=1..18); # Peter Luschny, Aug 13 2015

nmax = 18; a[n_] := Module[{e, dim = n*(nmax-1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0 , e[i, 0] = 0; For[k = i-1, k >= 0, k--, e[k, i-k] = e[k+1, i-k-1] + e[k, i-k-1] ], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i-k] = e[k-1, i-k+1] + e[k-1, i-k] ] ]]; e[0, 3*n]] ; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Jul 26 2013, after Maple *)

a(n) = (3*n)!*(1/(3*n)!-2/(n!*(2*n)!)+1/(n!)^3). - Peter Luschny, Aug 13 2015

A322938 a(n) = binomial(2*n + 2, n + 2) - 1.

Original entry on oeis.org

0, 3, 14, 55, 209, 791, 3002, 11439, 43757, 167959, 646645, 2496143, 9657699, 37442159, 145422674, 565722719, 2203961429, 8597496599, 33578000609, 131282408399, 513791607419, 2012616400079, 7890371113949, 30957699535775, 121548660036299, 477551179875951

Offset: 0

Peter Luschny, Feb 13 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001791, A014473, A030662 (d=0), A010763 (d=1), this sequence (d=2).

[Binomial(2*n+2,n+2) -1: n in [0..30]]; // G. C. Greubel, Apr 22 2024

aList := proc(len) local gf, ser; assume(Im(x) > 0);
gf := (2*x^2 - x + 1)/(2*(x - 1)*x^2) - (I*(2*x - 1))/(2*x^2*sqrt(4*x - 1));
ser := series(gf, x, len+4):
seq(coeff(ser, x, n), n=0..len) end: lprint(aList(25));

Table[Binomial[2 n + 2, n + 2] - 1, {n, 0, 25}]

[binomial(2*n+2,n+2) - 1 for n in range(31)] # G. C. Greubel, Apr 22 2024

Let G(x) = (2*x^2-x+1)/(2*(x-1)*x^2)-(I*(2*x-1))/(2*x^2*sqrt(4*x-1)) with Im(x) > 0, then a(n) = [x^n] G(x). The generating function G(x) satisfies the differential equation 9*x - 16*x^2 + 4*x^3 = (8*x^5 - 22*x^4 + 21*x^3 - 8*x^2 + x)*diff(G(x), x) + (12*x^4 - 36*x^3 + 38*x^2 - 16*x + 2)*G(x).
From Peter Bala, Feb 25 2022: (Start)
a(n) = Sum_{k = 0..n+1} binomial(n+k,k+1).
a(n) = Sum_{k = 0..n-1} binomial(n+k+2,k+1).
More generally, Sum_{k = 0..n+m} binomial(n+k,k+1) = Sum_{k = 0..n-1} binomial(n+k+m+1,k+1) = binomial(2*n+m+1,n) - 1. (End)
a(n) = A001791(n+1) - 1. - Hugo Pfoertner, Feb 26 2022
a(n) = n/(n+2) * binomial(2*n+2, n+1) * Sum_{k = 0..n+1} 1/binomial(n+k+1, k). - Peter Bala, Aug 05 2025

A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.

Original entry on oeis.org

1, 2, 3, 3, 6, 10, 4, 10, 20, 35, 5, 15, 35, 70, 126, 6, 21, 56, 126, 252, 462, 7, 28, 84, 210, 462, 924, 1716, 8, 36, 120, 330, 792, 1716, 3432, 6435, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310

Offset: 1

Daniel McLaury (daniel.mclaury(AT)gmail.com), Sep 11 2009

T(n,k) is the number of non-descending sequences with length k and last number is less or equal to n. T(n,k) is also the number of integer partitions (of any positive integer) with length k and largest part is less or equal to n. - Zlatko Damijanic, Dec 06 2024

			1;
2, 3;
3, 6, 10;
4, 10, 20, 35;
5, 15, 35, 70, 126;
6, 21, 56, 126, 252, 462;
7, 28, 84, 210, 462, 924, 1716;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A059481 with the first column (k = 0) removed.

Cf. A030662 (row sums), A001700 (diagonal).

Table[Binomial[n+k-1,k],{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 31 2021 *)

tabl(nn) = {for (n=1, nn, for (k=1, n, print1( binomial(n+k-1, k), ", ");); print(););} \\ Michel Marcus, Jun 12 2013

A331673 Sum of all base-n numbers with digit sum n and length exactly n.

Original entry on oeis.org

3, 79, 2299, 84361, 3872406, 216591677, 14378073683, 1107635176621, 97229999995138, 9583904327477305, 1048274845801847390, 126003010469828661807, 16510208629407273871884, 2342241434486480710216185, 357676630651821282153992579, 58498575553741083746904253333

Offset: 2

Alois P. Heinz, Feb 22 2020

The cardinality of these numbers is given by A030662(n-1)

			a(2) = 3 = 11_2.
a(3) = 79 = 11 + 13 + 15 + 19 + 21 = 102_3 + 111_3 + 120_3 + 201_3 + 210_3.

Alois P. Heinz, Table of n, a(n) for n = 2..200

Cf. A007953, A030662, A130835, A331672.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
      `if`(i=0, 0, add((p->[p[1], p[2]*k+p[1]*d])(
         b(n-d, i-1, k)), d=0..min(n, k-1))))
    end:
a:= n-> b(n$3)[2]-b(n, n-1, n)[2]:
seq(a(n), n=2..17);

A375694 Number A(n,k) of multiset permutations of {{1}^k, {2}^k, ..., {n}^k} with no fixed k-tuple {j}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 5, 2, 0, 1, 0, 19, 74, 9, 0, 1, 0, 69, 1622, 2193, 44, 0, 1, 0, 251, 34442, 362997, 101644, 265, 0, 1, 0, 923, 756002, 62924817, 166336604, 6840085, 1854, 0, 1, 0, 3431, 17150366, 11729719509, 305225265804, 136221590695, 630985830, 14833, 0

Offset: 0

Alois P. Heinz, Aug 24 2024

			A(2,2) = 5: 1212, 1221, 2112, 2121, 2211.
A(2,3) = 19: 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
A(3,2) = 74: 121323, 121332, 122313, 122331, 123123, 123132, 123213, 123231, 123312, 123321, 131223, 131232, 131322, 132123, 132132, 132312, 132321, 133122, 133212, 133221, 211323, 211332, 212313, 212331, 213123, 213132, 213213, 213231, 213312, 213321, 221313, 221331, 223113, 223131, 223311, 231123, 231132, 231213, 231231, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311223, 311232, 311322, 312123, 312132, 312312, 312321, 313122, 313212, 313221, 321123, 321132, 321213, 321231, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121.
A(4,1) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Square array A(n,k) begins:
  1,  1,      1,         1,            1,               1, ...
  0,  0,      0,         0,            0,               0, ...
  0,  1,      5,        19,           69,             251, ...
  0,  2,     74,      1622,        34442,          756002, ...
  0,  9,   2193,    362997,     62924817,     11729719509, ...
  0, 44, 101644, 166336604, 305225265804, 623302086965044, ...

Alois P. Heinz, Antidiagonals n = 0..53, flattened

Columns k=0-2 give: A000007, A000166, A374980.
Rows n=0-2 give: A000012, A000004, A030662.
Main diagonal gives A375693.
Cf. A060538, A089759, A187783, A372307.

A:= (n, k)-> add((-1)^(n-j)*binomial(n, j)*(k*j)!/k!^j, j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(k*j)!/k!^j.

A237664 Interpolation polynomial through n+1 points (0,1), (1,1), ..., (n-1,1) and (n,n) evaluated at 2n.

Original entry on oeis.org

0, 1, 7, 41, 211, 1009, 4621, 20593, 90091, 388961, 1662805, 7054321, 29745717, 124807201, 521515801, 2171645281, 9016205851, 37337699521, 154277300101, 636214748401, 2619084047581, 10765157488801, 44186078238121, 181135476007201, 741694884711301

Offset: 0

Alois P. Heinz, Feb 11 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000290 (evaluated at n+1), A127736 (at n+2), A237622 (n points).

a:= proc(n) option remember; `if`(n<2, n,
       ((n-1)*(3*n-4)*(5*n-3) *a(n-1)
        -2*(2*n-3)*(3*n^2-4*n+2) *a(n-2))/
        (n*(3*n^2-10*n+9)))
    end:
seq(a(n), n=0..30);

CoefficientList[Series[(6*x-1)/Sqrt[1-4*x]^3-1/(x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 14 2014 *)
a[n_] := Module[{m}, InterpolatingPolynomial[Table[{k, If[k == n, n, 1]}, {k, 0, n}], m] /. m -> 2n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020 *)

G.f.: (6*x-1)/sqrt(1-4*x)^3 - 1/(x-1).
a(n) ~ sqrt(n)*4^n/sqrt(Pi). - Vaclav Kotesovec, Feb 14 2014
From Gregory Morse, Mar 19 2021: (Start)
a(n) = (2*n)!*(n-1)/(n!)^2 + 1.
a(n) = A030662(n-1)*(n-1) + n, for n > 0. (End)
E.g.f.: exp(x) * (1 - exp(x) * ((1 - 2*x) * BesselI(0,2*x) - 2 * x * BesselI(1,2*x))). - Ilya Gutkovskiy, Nov 19 2021

cp's OEIS Frontend

A308292 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!*j!*k!*l!).

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Maple

Mathematica

PARI

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A181991 n-alternating permutations of length 4n.

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Maple

Mathematica

Formula

A206735 Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A211213 n-alternating permutations of length 3n.

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Maple

Mathematica

Formula

A322938 a(n) = binomial(2*n + 2, n + 2) - 1.

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Magma

Maple

Mathematica

SageMath

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A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.

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Mathematica

PARI

A331673 Sum of all base-n numbers with digit sum n and length exactly n.

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A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!j!k!*l!).