A060354 -id:A060354 - cp's OEIS frontend

A294958 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*((k-2)^2+k)/2).

1, 1, 3, 9, 28, 75, 198, 494, 1243, 3061, 7500, 18055, 43057, 101292, 236178, 545218, 1248480, 2835059, 6390360, 14298631, 31778782, 70168935, 153993321, 335977369, 728962258, 1573189113, 3377881482, 7217395643, 15348900996, 32494548816, 68494383520, 143773075158, 300568066729

Offset: 0

Ilya Gutkovskiy, Nov 12 2017

nonn

Euler transform of A060354.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A057145, A060354, A294959.

N:=100:
S:= series(mul(1/(1 - x^k)^(k*((k-2)^2+k)/2),k=1..N),x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Nov 12 2017

nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A060354(k).

a(n) ~ exp(2*Zeta'(-1) + 3*Zeta(3) / (8*Pi^2) - Pi^16 / (1036800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - 2*Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3)/2 + (-Pi^12 / (3600000 * 2^(2/5) * 3^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (150 * 2^(2/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(4/5) * 3^(2/5) * Zeta(5)^(7/5)) + 2^(1/5) * Zeta(3) / (3*Zeta(5))^(2/5)) * n^(2/5) - (Pi^4 / (60 * 2^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5*(3*Zeta(5))^(1/5) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(53/400) / (2^(47/200) * sqrt(5*Pi) * n^(253/400)). - Vaclav Kotesovec, Nov 12 2017

A294959 Expansion of Product_{k>=1} (1 + x^k)^(k*((k-2)^2+k)/2).

Original entry on oeis.org

1, 1, 2, 8, 23, 64, 160, 397, 968, 2372, 5714, 13617, 32007, 74396, 171222, 390629, 883922, 1984631, 4423528, 9790146, 21524829, 47027558, 102135967, 220565018, 473743833, 1012274948, 2152271718, 4554344649, 9593260912, 20118418061, 42012556671, 87375161720, 181001416773

Offset: 0

Ilya Gutkovskiy, Nov 12 2017

nonn

Weigh transform of A060354.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A057145, A060354, A294958.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(((i-2)^2+i)*i/2, j)*g(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 12 2017

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} (1 + x^k)^A060354(k).

a(n) ~ exp(-2401 * Pi^16 / (3499200000000 * Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - 2*Zeta(3)^2 / (25*Zeta(5)) + (-343*Pi^12 / (810000000 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (360000 * 2^(1/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + (3/2)^(1/5) * Zeta(3) / (5*Zeta(5))^(2/5)) * n^(2/5) - (7*Pi^4 / (180 * 2^(4/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(12/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(69/80) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 12 2017

A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).

Original entry on oeis.org

1, 1, 3, 6, 5, 95, 336, 2597, 26832, 197577, 1847800, 14621101, 129754956, 1146534701, 12342194879, 161225146370, 2464561564936, 39642413790129, 620059254486798, 9430493858327959, 136438759335452360, 1881721996407396801, 24999081626667425376, 321601467988647184779

Offset: 0

Ilya Gutkovskiy, Oct 03 2020

nonn

			a(3) = 6 because the third triangular number is 6 and we have [6, 0, 0], [0, 6, 0], [0, 0, 6], [3, 3, 0], [3, 0, 3] and [0, 3, 3].

Eric Weisstein's World of Mathematics, Polygonal Number
Index to sequences related to polygonal numbers

Cf. A060354, A088218, A196010, A298329, A335634, A337762, A337763, A337764.

Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 0, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 0, 23}]

p(n,k) = {k * (k * (n - 2) - n + 4) / 2}
a(n) = {my(m=p(n,n)); polcoef((sum(k=0, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ Andrew Howroyd, Oct 03 2020

a(n) = [x^p(n,n)] (Sum_{k=0..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A335634 Number of ordered ways of writing the n-th n-gonal number as a sum of n nonzero n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 0, 1, 30, 180, 700, 3780, 11844, 50610, 325820, 5803380, 126594910, 2114901789, 28282722650, 323420067880, 3190581939996, 29336527986960, 245438739897312, 1967485926594030, 16000631392009320, 184418174847183508, 4054670001158799616, 111835386569787369559

Offset: 0

Ilya Gutkovskiy, Oct 03 2020

nonn

			a(4) = 1 because the fourth square is 16 and we have [4, 4, 4, 4].

Eric Weisstein's World of Mathematics, Polygonal Number
Index to sequences related to polygonal numbers

Cf. A060354, A298330, A298858, A335633, A337762, A337763, A337764.

Join[{1}, Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 1, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 1, 24}]]

p(n,k) = {k * (k * (n - 2) - n + 4) / 2}
a(n) = {my(m=p(n,n)); polcoef((sum(k=1, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ Andrew Howroyd, Oct 03 2020

a(n) = [x^p(n,n)] (Sum_{k=1..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A384243 a(n) = 2^(n-6)n(n^3 - 6n^2 + 19n - 14).

Original entry on oeis.org

0, 0, 1, 6, 30, 140, 600, 2352, 8512, 28800, 92160, 281600, 827904, 2356224, 6522880, 17633280, 46694400, 121438208, 310837248, 784465920, 1954938880, 4816896000, 11747721216, 28386000896, 68010639360, 161690419200, 381681664000, 895098028032, 2086448136192, 4836200284160

Offset: 0

Enrique Navarrete, May 23 2025

a(n) is the number of strings of length n defined on {0, 1, 2, 3} that have exactly two 2's, zero or two 3's, and have no restriction on the number of 0's and 1's.

			a(4) = 30 since the strings are the 6 permutations of 2233, the 6 permutations of 1122, the 6 permutations of 0022, and the 12 permutations of 0122.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A060354, A327319, A383778.

A384243[n_] := 2^(n-6)*n*(n-1)*(n*(n-5)+14); Array[A384243, 30, 0] (* or *)
LinearRecurrence[{10, -40, 80, -80, 32}, {0, 0, 1, 6, 30}, 30] (* Paolo Xausa, May 27 2025 *)

E.g.f.: exp(2*x)*(x^2/2 + x^4/4).

G.f.: x^2*(1 - 4*x + 10*x^2)/(1 - 2*x)^5. - Stefano Spezia, May 23 2025

A122061 First pentagonal number, 2nd hexagonal number, 3rd heptagonal number, 4th octagonal number and then repeat the same pattern: 5th pentagonal, 6th hexagonal, 7th heptagonal, 8th octagonal, etc.

Original entry on oeis.org

1, 6, 18, 40, 35, 66, 112, 176, 117, 190, 286, 408, 247, 378, 540, 736, 425, 630, 874, 1160, 651, 946, 1288, 1680, 925, 1326, 1782, 2296, 1247, 1770, 2356, 3008, 1617, 2278, 3010, 3816, 2035, 2850, 3744, 4720, 2501, 3486, 4558, 5720, 3015, 4186, 5452

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 14 2006

nonn

From a quiz.

A. Wareham, Test Your Brain Power, Ward Lock Ltd (1995).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1).

Cf. A060354.

fn[n_]:=Module[{r=Mod[n,4]},Which[r==1,(n(3n-1))/2,r==2,(n(4n-2))/2,r==3,(n(5n-3))/2,r==0,(n(6n-4))/2]]; Array[fn,50] (* or *) LinearRecurrence[ {0,0,0,3,0,0,0,-3,0,0,0,1},{1,6,18,40,35,66,112,176,117,190,286,408},50] (* Harvey P. Dale, Mar 01 2015 *)

for(n=1,60,m=(n+3)%4;print1(n*((m+3)*n-m-1)/2,","))

a(n) = n*(3*n-1)/2 if n=1 mod 4 or n*(4*n-2)/2 if n=2 mod 4 or n*(5*n-3)/2 if n=3 mod 4 or n*(6*n-4)/2 if n=0 mod 4

a(n)=3*a(n-4)-3*a(n-8)+a(n-12) for n>11. - Harvey P. Dale, Mar 01 2015

A129699 Least nonnegative m such that P(n+3,n) + P(n+3,m) is prime where P(k,n) is n-th k-gonal number, or -1 if no such value exists.

Original entry on oeis.org

2, 1, 0, 4, 2, 2, 4, 4, 2, 4, 3, 6, 73, 4, 3, 16, 9, 6, 7, 2, 17, 10, 3, 2, 10, 2, 36, 58, 9, 2, 7, 4, 6, 82, 3, 2, 25, 4, 11, 10, 2, 6, 43, 2, 14, 46, 11, 38, 37, 2, 32, 130, 14, 2, 28, 2, 5, 28, 4, 14, 37, 16, 24, 16, 2, 2, 40, 4, 2, 10, 8, 6, 46, 22, 3, 28, 5, 18, 16, 2, 26, 10, 19, 12, 10, 8

Offset: 0

Jonathan Vos Post, Jun 01 2007

Define array A[k,n] for k>2, n>=0, where A[k,n] = n-th k-gonal number = k((n-2)*k - (n-4))/2. Then define array B[k,n] = least m such that the A[k,n] + m-th k-gonal number is prime. This sequence is the main diagonal of B. The array B[k,n] begins: k.|.B[k,n] 3.|..2.1.0.1.1.7.4.1.1.7.3... 4.|.-1.2.1.2.1.2.5.2.3.4.1... 5.|..2.3.0.1.1.3.4.1.4.3... 6.|.-1.1.1.4.1.4.4.2.7.4... 7.|..2.3.0.1.2.3.7.1.1.4... 8.|.-1.4.3.2.1.2.1.4.3.2... B[4,0] = -1 because 0th 4-gonal number is 0th square = 0 and 0 + c^2 cannot be prime for any integer c. B[5,6] = 4 because 6th + 5th pentagonal numbers = 51 + 22 = 73 is prime. B[8,2] = 3 because 3rd + 2nd octagonal numbers = 21 + 8 = 29 is prime.

The sequence of associated primes starts 3, 2, 5, 43, 41, 73, 157, 227, 271, 433, 541, 857, 35107, 1193, 1427,... - R. J. Mathar, Jun 12 2008

Eric Weisstein's World of Mathematics, Polygonal Number.

Cf. A000040, A000217, A060354.

P := proc(k,n) n/2*((k-2)*n-k+4) ; end: A129699 := proc(n) for m from 0 to 100000 do if isprime(P(n+3,n)+P(n+3,m)) then RETURN(m) ; fi ; od: RETURN(-1) ; end: for n from 0 to 200 do printf("%d,",A129699(n)) ; od: # R. J. Mathar, Jun 12 2008

a(n) = min{m: m-th (n+3)-gonal number + n-th (n+3)-gonal number is prime}.

Corrected and extended by R. J. Mathar, Jun 12 2008

A228399 The number of permutations of length n sortable by 2 cut-and-paste moves.

Original entry on oeis.org

1, 2, 6, 24, 120, 577, 2208, 6768, 17469, 39603, 81272, 154225, 274802, 464985, 753556, 1177362, 1782687, 2626731, 3779196, 5323979, 7360972, 10007969, 13402680, 17704852, 23098497, 29794227, 38031696, 48082149, 60251078, 74880985, 92354252, 113096118

Offset: 1

Vincent Vatter, Aug 21 2013

			The shortest permutations which cannot be sorted by 2 cut-and-paste moves are of length 6.

D. W. Cranston, I. H. Sudborough, and D. B. West, Short proofs for cut-and-paste sorting of permutations, Discrete Math. 307, 22 (2007), 2866-2870.
C. Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657, 2014.
C. Homberger, V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946, 2013.

Cf. A060354, A228400.

CoefficientList[Series[(1/x) (-1 + (x^11 - 5 x^10 - 2 x^9 + 44 x^8 - 23 x^7 - 87 x^6 - 22 x^5 - 24 x^4 + 22 x^3 - 16 x^2 + 6 x - 1)/(x - 1)^7), {x,
   0, 40}], x] (* Bruno Berselli, Aug 22 2013 *)

G.f.: -1 + (x^11 - 5*x^10 - 2*x^9 + 44*x^8 - 23*x^7 - 87*x^6 - 22*x^5 - 24*x^4 + 22*x^3 - 16*x^2 + 6*x - 1)/(x - 1)^7.

a(n) = n! for 0 < n < 5; for n > 4, a(n) = -18 + n*(107*n^5 -1077*n^4 +2225*n^3 +12345*n^2 -61732*n +80532)/720. [Bruno Berselli, Aug 22 2013]

A228400 The number of permutations of length n sortable by 3 cut-and-paste moves.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 36757, 223898, 1055479, 3973264, 12530496, 34434065, 84883448, 191729212, 403095882, 798248632, 1502630530, 2708156958, 4700026333, 7891491375, 12868232903, 20444188490, 31730911273, 48222769794, 71900547943

Offset: 1

Vincent Vatter, Aug 21 2013

			The shortest permutations that cannot be sorted by 3 cut-and-paste moves are of length 8.

D. W. Cranston, I. H. Sudborough, and D. B. West, Short proofs for cut-and-paste sorting of permutations, Discrete Math. 307, 22 (2007), 2866-2870.
C. Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657, 2014.
C. Homberger, V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946, 2013.

Cf. A060354, A228399.

CoefficientList[Series[-(2 x^19 + 28 x^17 - 90 x^16 + 31 x^15 - 329 x^14 + 2874 x^13 - 1487 x^12 - 13363 x^11 + 17425 x^10 + 8876 x^9 - 16945*x^8 - 8185 x^7 - 1326 x^6 - 48 x^5 - 120 x^4 + 66 x^3 - 31*x^2 + 8*x - 1)/(x - 1)^10, {x, 0, 40}], x] (* Bruno Berselli, Aug 23 2013 *)

G.f.: -x*(2*x^19 + 28*x^17 - 90*x^16 + 31*x^15 - 329*x^14 + 2874*x^13 - 1487*x^12 - 13363*x^11 + 17425*x^10 + 8876*x^9 - 16945*x^8 - 8185*x^7 - 1326*x^6 - 48*x^5 - 120*x^4 + 66*x^3 - 31*x^2 + 8*x - 1)/(x - 1)^10

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

cp's OEIS Frontend

A294958 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*((k-2)^2+k)/2).

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A294959 Expansion of Product_{k>=1} (1 + x^k)^(k*((k-2)^2+k)/2).

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A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).

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A335634 Number of ordered ways of writing the n-th n-gonal number as a sum of n nonzero n-gonal numbers.

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A384243 a(n) = 2^(n-6)*n*(n^3 - 6*n^2 + 19*n - 14).

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A122061 First pentagonal number, 2nd hexagonal number, 3rd heptagonal number, 4th octagonal number and then repeat the same pattern: 5th pentagonal, 6th hexagonal, 7th heptagonal, 8th octagonal, etc.

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A129699 Least nonnegative m such that P(n+3,n) + P(n+3,m) is prime where P(k,n) is n-th k-gonal number, or -1 if no such value exists.

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A228399 The number of permutations of length n sortable by 2 cut-and-paste moves.

A384243 a(n) = 2^(n-6)n(n^3 - 6n^2 + 19n - 14).