A006472 -id:A006472 - cp's OEIS frontend

A334130 Numbers that can be written as a product of distinct triangular numbers.

0, 1, 3, 6, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 180, 190, 198, 210, 216, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 450

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A006472, A054731, A085780, A085782, A334129.

N:= 1000: # for all terms <= N
S:= {0,1}:
for i from 2 do
  t:= i*(i+1)/2;
  if t > N then break fi;
  S:= S union select(`<=`,map(`*`,S,t),N)
od:
sort(convert(S,list)); # Robert Israel, Apr 21 2020

A367574 Decimal expansion of BesselI(0,2*sqrt(2)).

Original entry on oeis.org

4, 2, 5, 2, 3, 5, 0, 8, 7, 9, 5, 0, 2, 6, 2, 3, 8, 2, 5, 2, 9, 3, 2, 3, 0, 8, 2, 4, 0, 8, 9, 5, 1, 0, 3, 0, 2, 1, 0, 7, 6, 8, 6, 0, 9, 8, 2, 2, 7, 0, 6, 7, 5, 3, 6, 4, 4, 7, 4, 3, 2, 1, 9, 9, 9, 6, 9, 3, 7, 7, 7, 8, 1, 1, 3, 0, 4, 2, 0, 6, 4, 4, 7, 8, 7, 0, 3, 8, 6, 7, 2, 8, 0, 1, 8, 9, 7, 2, 8, 1

Offset: 1

Ilya Gutkovskiy, Nov 28 2023

			4.252350879502623825293230824089510302...

Cf. A000217, A006472, A070910, A072334, A367709, A367710.

RealDigits[BesselI[0, 2 Sqrt[2]], 10, 100][[1]]

Equals Sum_{k>=0} 2^k / k!^2.

Equals Sum_{i>=0} (i+1)/Product_{j=1..i} A000217(j). - Davide Rotondo, Feb 25 2025

A375837 Triangle read by rows: T(n,k) is the number of rooted chains starting with the cycle (1)(2)(3)...(n) of length k of permutation poset of n letters.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 3, 0, 1, 23, 41, 18, 0, 1, 119, 455, 515, 180, 0, 1, 719, 5139, 10985, 9255, 2700, 0, 1, 5039, 62713, 222551, 334040, 225855, 56700, 0, 1, 40319, 840265, 4619447, 10899840, 12686030, 7193340, 1587600, 0, 1, 362879, 12383329, 101128653, 350413245, 620801580, 592261110, 289918440, 57153600

Offset: 0

Rajesh Kumar Mohapatra, Ranjan Kumar Dhani, and Subhashree Sahoo, Aug 31 2024

			Triangle T(n,k) begins:
  n\k | 0  1   2     3     4      5      6     7 ...
 -----+-----------------------------------------
  0   | 1;
  1   | 0, 1;
  2   | 0, 1, 1;
  3   | 0, 1, 5, 3;
  4   | 0, 1, 23, 41, 18;
  5   | 0, 1, 119, 455, 515, 180;
  6   | 0, 1, 719, 5139, 10985, 9255, 2700;
  7   | 0, 1, 5039, 62713, 222551, 334040, 225855, 56700;
  ...
The T(3, 2) = 5 chains in the poset of the permutations of {1, 2, 3} are: {(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132)}.

Sean A. Irvine, Java program (github)

Cf. A000007 (column k=0), A057427 (column k=1), A006472 (diagonal), A375838 (row sums).

Cf. A048994, A331955, A330804, A331956, A331957, A375835, A375836.

T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[(n == 0 && k == 0) || k == 1, 1, Sum[If[r >= 0, Abs[StirlingS1[n, r]]*T[r, k - 1], 0], {r, k - 1, n - 1}]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* corrected Jul 01 2025 *)

Let Stirling1(n, k) denote the unsigned Stirling numbers of the first kind (A132393).
T(0, 0) = 1, T(0, k) = 0 for k > 0 and T(n, 1) = 1 for n > 1.
T(n, k) = Sum_{i_(k-1)=k-1..n-1} (Sum_{i_(k-2)=k-2..i_(k-1) - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling1(n,i_(k-1)) * Stirling1(i_(k-1),i_(k-2)) * ... * Stirling1(i_3,i_2) * Stirling1(i_2,i_1)))...)), where 2 <= k <= n.

A386710 Decimal expansion of BesselI(2, 2 * sqrt(2)).

Original entry on oeis.org

1, 8, 5, 7, 5, 1, 7, 7, 8, 0, 2, 2, 9, 2, 1, 9, 1, 0, 8, 7, 7, 0, 5, 9, 8, 1, 8, 7, 6, 5, 3, 1, 3, 7, 1, 5, 0, 1, 3, 9, 0, 4, 9, 0, 9, 1, 1, 3, 5, 7, 0, 5, 0, 9, 1, 7, 2, 0, 6, 3, 4, 6, 3, 6, 1, 1, 2, 0, 8, 2, 2, 2, 1, 6, 4, 7, 0, 8, 7, 1, 8, 0, 7, 3, 4, 5, 8, 2, 7, 2, 2, 8, 3, 0, 6, 6, 5, 7, 7, 3, 9, 3, 8, 5, 8

Offset: 1

Kelvin Voskuijl, Jul 30 2025

			1.8575177802292191087705981876531371501390490911357...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A386927 (continued fraction).
Cf. A096789 (for factorial squared).
Cf. A006472 (triangular polygorials).

RealDigits[BesselI[2, 2 * Sqrt[2]],10,100][[1]] (* Stefano Spezia, Aug 02 2025 *)

besseli(2, 2*sqrt(2)) \\ Amiram Eldar, Aug 02 2025

Equals Sum_{k >= 1} 2^k/((k-1)!*(k+1)!).

A006473 a(n) = binomial(n,2)!/n!.

Original entry on oeis.org

1, 30, 30240, 1816214400, 10137091700736000, 7561714896123855667200000, 1025113885554181044609786839040000000, 32964677266721834921175915315161407370035200000000, 318071672921132854486459356650996997744817246158245068800000000000

Offset: 3

N. J. A. Sloane

nonn

a(n) is also the number of distinct possible (n-1)-dimensional simplices if the (n-1)*n/2 1-faces are given (up to symmetry, rotation, reflection). - Dan Dima, Nov 03 2011

a(n) is also the number of edge labelings of the complete graph on n vertices. - Nikos Apostolakis, Jul 09 2013

			a(3)=1 since there is one possible triangle if the 3 edges are given and a(4)=30 since there are 30 distinct possible tetrahedra if the 6 edges are given. - _Dan Dima_, Nov 03 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 3..30
O. Frank and K. Svensson, On probability distributions of single-linkage dendrograms, Journal of Statistical Computation and Simulation, 12 (1981), 121-131. (Annotated scanned copy)
C. L. Mallows, Note to N. J. A. Sloane circa 1979.

Table[Binomial[n,2]!/n!,{n,3,20}] (* Harvey P. Dale, May 08 2013 *)

A061213 a(n) = 1 + product of first n positive triangular numbers (A000217).

Original entry on oeis.org

2, 4, 19, 181, 2701, 56701, 1587601, 57153601, 2571912001, 141455160001, 9336040560001, 728211163680001, 66267215894880001, 6958057668962400001, 834966920275488000001, 113555501157466368000001

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

			a(6) = 1 * 3 * 6 * 10 *15 * 21 + 1 = 56701.

Harry J. Smith, Table of n, a(n) for n=1..100

Cf. A000217, A006472.

for n from 1 to 30 do printf(`%d,`,1+product(j*(j+1)/2, j=1..n)) od:

a(n) = { 1 + prod(k=1, n, k*(k + 1)/2) } \\ Harry J. Smith, Jul 19 2009

a(n) = A006472(n+1) + 1.

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

More terms from James Sellers, Apr 24 2001

Offset changed from 0 to 1 and example corrected by Harry J. Smith, Jul 19 2009

A139769 T(n,k) = [x^k] Product_{m=1..n} d/dx Sum_{i=1..m} x^i; triangle read by rows, n >= 0, 0 <= k <= A161680(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 7, 6, 1, 6, 18, 36, 49, 46, 24, 1, 8, 33, 94, 204, 354, 497, 562, 501, 326, 120, 1, 10, 52, 188, 528, 1222, 2406, 4102, 6116, 7996, 9132, 9014, 7541, 5116, 2556, 720, 1, 12, 75, 326, 1105, 3106, 7513, 16014, 30558, 52752, 82938, 119230, 156983

Offset: 0

Roger L. Bagula, Jun 13 2008

Row sums are A006472(n+1).

T(n, binomial(n,2)-k) is the number of rank-k intervals in the middle order on permutations. (See Bouvel et al. reference.) - Bridget Tenner, May 24 2024

			Triangle T(n,k) begins:
  1;
  1;
  1, 2;
  1, 4,  7,  6;
  1, 6, 18, 36,  49,  46,  24;
  1, 8, 33, 94, 204, 354, 497, 562, 501, 326, 120;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Mathilde Bouvel, Luca Ferrari, and Bridget Eileen Tenner, Between weak and Bruhat: the middle order on permutations, arXiv:2405.08943 [math.CO], 2024.

Cf. A000142, A008302 (Mahonian numbers), A006472, A010551, A161680, A259459.

a := Table[CoefficientList[Product[Sum[D[x^i, x], {i, 1, m}], {m, 1, n}], x], {n, 0, 7}]; Flatten[a]

From Alois P. Heinz, May 24 2024: (Start)
|Sum_{k=0..binomial(n,2)} (-1)^k T(n,k)| = A010551(n).
Sum_{k=0..binomial(n,2)} (binomial(n,2)-k)*T(n,k) = A259459(n-2) for n>=2. (End)

Edited by Alois P. Heinz, May 24 2024

A193439 exp( Sum_{n>=1} a(n-1)*x^n/n!^3 ) = Sum_{n>=0} a(n)/n!^3.

Original entry on oeis.org

1, 1, 5, 68, 1936, 99336, 8326912, 1063584640, 196475565312, 50403792222720, 17382740425346304, 7847087503671023616, 4535069738055660564480, 3292828639234241171484672, 2955617286961757422869504000, 3233957295970672142211481337856

Offset: 0

Paul D. Hanna, Jul 25 2011

nonn

Compare to: exp(Sum_{n>=1} A006472(n)*x^n/n!^2) = Sum_{n>=0} A006472(n+1)/n!^2 where A006472(n) = n!*(n-1)!/2^(n-1).

			E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/(n!)^^3 starts as
A(x) = 1 + x + 5*x^2/2!^3 + 68*x^3/3!^3 + 1936*x^4/4!^3 + 99336*x^5/5!^3 + 8326912*x^6/6!^3 + 1063584640*x^7/7!^3 + 196475565312*x^8/8!^3 + 50403792222720*x^9/9!^3 + 17382740425346304*x^10/10!^3 + 7847087503671023616*x^11/11!^3 +...+ a(n)*x^n/n!^3 +...
where
log(A(x)) = x + x^2/2!^3 + 5*x^3/3!^3 + 68*x^4/4!^3 + 1936*x^5/5!^3 + 99336*x^6/6!^3 + 8326912*x^7/7!^3 +...+ a(n-1)*x^n/n!^3 +...
As a power series in x with reduced fractional coefficients, the e.g.f. begins
A(x) = 1 + x + 5/8*x^2 + 17/54*x^3 + 121/864*x^4 + 4139/72000*x^5 + 32527/1458000*x^6 + 1661851/200037600*x^7 + 85275853/28449792000*x^8 + 729221531/691329945600*x^9 + 2514864066167/6913299456000000*x^10 + 141910581301921/1150200196992000000*x^11 + 8201442668648113/198754594040217600000*x^12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..299

{a(n) = n!^3*polcoeff( exp(x+sum(m=2,n,a(m-1)*x^m/m!^3+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1); for(i=0, n, A = exp( intformal(1/x*intformal(1/x*intformal(A +x*O(x^n)))))); (n!)^3*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 30 2019

E.g.f.: A(x) = exp( Integral 1/x * Integral 1/x * Integral A(x) dx dx dx ). - Paul D. Hanna, Apr 30 2019

A203157 (n-1)-st elementary symmetric function of the first n triangular numbers.

Original entry on oeis.org

1, 4, 27, 288, 4500, 97200, 2778300, 101606400, 4629441600, 257191200000, 17116074360000, 1344389840640000, 123067686661920000, 12988374315396480000, 1565562975516540000000, 213751531590524928000000, 32817539834507780352000000

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

			Let esf abbreviate "elementary symmetric function".  Then
0th esf of {1}:  1
1st esf of {1,3}:  1+3=4
2nd esf of {1,3,6} is 1*3+1*6+3*6=27

Cf. A000217, A006472 (n-th symm. func.), A000292 (1st symm. func.).

f[k_] := k (k + 1)/2; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}]  (* A203157 *)

Conjecture: 2*(-n+1)*a(n) +n^3*a(n-1)=0. - R. J. Mathar, Oct 01 2016

A233005 a(n) = floor(Pt(n)/n!), where Pt(n) is product of first n positive triangular numbers (A000217).

Original entry on oeis.org

1, 1, 3, 7, 22, 78, 315, 1417, 7087, 38981, 233887, 1520268, 10641881, 79814109, 638512875, 5427359437, 48846234937, 464039231906, 4640392319062, 48724119350156, 535965312851718, 6163601097794765, 73963213173537187, 924540164669214843, 12019022140699792968

Offset: 1

Alex Ratushnyak, Dec 03 2013

nonn

			a(4) = 7, because, the first four triangular numbers being 1, 3, 6, 10, their product is 180, which divided by 4! is 15/2 = 7.5.
a(5) = 22, because, the first five triangular numbers being 1, 3, 6, 10, 15, their product is 2700, which divided by 5! is 45/2 = 22.5.

Harvey P. Dale, Table of n, a(n) for n = 1..505

Cf. A000142, A000217.
Cf. A006472 (triangular factorial, essentially equal to Pt(n)).
Cf. A067667 (Pt(n)/n! for n's of the form 2^k-1).

With[{nn=30},Floor[#[[1]]/#[[2]]]&/@Thread[{FoldList[Times,Accumulate[ Range[ nn]]],Range[nn]!}]] (* Harvey P. Dale, Apr 02 2017 *)

f=t=1
for n in range(1,33):
  t*=n*(n+1)//2
  f*=n
  print(t//f, end=', ')

a(n) = floor((n+1)!/2^n). - Yifan Xie, Mar 05 2023

cp's OEIS Frontend

A334130 Numbers that can be written as a product of distinct triangular numbers.

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A367574 Decimal expansion of BesselI(0,2*sqrt(2)).

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A375837 Triangle read by rows: T(n,k) is the number of rooted chains starting with the cycle (1)(2)(3)...(n) of length k of permutation poset of n letters.

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A386710 Decimal expansion of BesselI(2, 2 * sqrt(2)).

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A006473 a(n) = binomial(n,2)!/n!.

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A061213 a(n) = 1 + product of first n positive triangular numbers (A000217).

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A139769 T(n,k) = [x^k] Product_{m=1..n} d/dx Sum_{i=1..m} x^i; triangle read by rows, n >= 0, 0 <= k <= A161680(n).

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A193439 exp( Sum_{n>=1} a(n-1)*x^n/n!^3 ) = Sum_{n>=0} a(n)/n!^3.

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