A132317 -id:A132317 - cp's OEIS frontend

A132318 Triangle, read by rows, where T(n,k) = [x^(k*2^(n-1))] Product_{i=0..n-1} (1 + x^(2^i))^(2^(n-i-1)) for n>0 with T(0,0)=1.

1, 1, 1, 1, 2, 1, 1, 15, 15, 1, 1, 1024, 2046, 1024, 1, 1, 7048181, 60060682, 60060682, 7048181, 1, 1, 469389728563470, 72057594037927935, 143176408618728932, 72057594037927935, 469389728563470, 1, 1, 2954306864416502250656677496683

Offset: 0

Paul D. Hanna, Aug 19 2007

There are n*2^(n-1)+1 coefficients in P(n) = Product_{i=0..n-1} (1 + x^(2^i))^(2^(n-i-1)) for n>0; in this triangle, row n consists of coefficients of x^(k*2^(n-1)) in P(n) as k=0..n.

			Triangle begins:
1;
1,1;
1,2,1;
1,15,15,1;
1,1024,2046,1024,1;
1,7048181,60060682,60060682,7048181,1;
1,469389728563470,72057594037927935,143176408618728932,72057594037927935,469389728563470,1;
Examples:
T(2,1) = [x^(1*2)] (1+x)^2*(1+x^2) = 2;
T(3,1) = [x^(1*4)] (1+x)^4*(1+x^2)^2*(1+x^4) = 15;
T(4,3) = [x^(3*8)] (1+x)^8*(1+x^2)^4*(1+x^4)^2*(1+x^8) = 1024;
T(5,3) = [x^(3*16)] (1+x)^16*(1+x^2)^8*(1+x^4)^4*(1+x^8)^2*(1+x^16) = 60060682.

Eric Weisstein, Mathworld, Series Multisection.

Cf. A132317 (column 1), A132316.

{T(n,k)=if(n==0,1,polcoeff(prod(i=0,n-1,(1+x^(2^i)+x*O(x^(k*2^(n-1))))^(2^(n-i-1))),k*2^(n-1)))}

Row sums equal 2^(2^n - n) for n>0 - improved formula and proof by Max Alekseyev, Aug 19 2007.

A132316 a(n) = [x^n] Product_{i=0..n} (1 + x^(2^i) )^(2^(n-i)).

Original entry on oeis.org

1, 2, 8, 88, 2812, 284832, 96344064, 112162777984, 458279216351168, 6667184111642112512, 349410072608155198029824, 66605152356815910201401874432, 46557942811582437260863430233248768

Offset: 0

Paul D. Hanna, Aug 18 2007

nonn

			a(2) = [x^2] (1+x)^4*(1+x^2)^2*(1+x^4) = 8;
a(3) = [x^3] (1+x)^8*(1+x^2)^4*(1+x^4)^2*(1+x^8) = 88;
a(4) = [x^4] (1+x)^16*(1+x^2)^8*(1+x^4)^4*(1+x^8)^2*(1+x^16) = 2812.

Cf. A132317, A132318.

Table[SeriesCoefficient[Product[(1 + x^(2^j))^(2^(n-j)),{j,0,n}],{x,0,n}], {n,0,15}] (* Vaclav Kotesovec, Oct 09 2020 *)

{a(n)=polcoeff(prod(i=0,#binary(n),(1 + x^(2^i) +x*O(x^n))^(2^(n-i))), n)}

a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Oct 09 2020

A175631 a(n) = (n-th pentagonal number) modulo (n-th triangular number).

Original entry on oeis.org

0, 2, 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377, 1430

Offset: 1

Zak Seidov, Jul 29 2010

nonn

			a(1)=0 because (1(3-1)/2) mod (1(1+1)/2) = 1 mod 1 = 0,
a(2)=2 because (2(6-1)/2) mod (2(2+1)/2) = 5 mod 3 = 2.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096 (n(n+3)/2), A000217 (triangular numbers), A000326 (pentagonal numbers), A175630 (n-th pentagonal number mod (n+2)).

Cf. A080956, A132315, A132316, A132317.

[n lt 4 select 1+(-1)^n else n*(n-3)/2: n in [1..60]]; // G. C. Greubel, Jan 30 2022

Table[Mod[n(3n-1)/2, n(n+1)/2],{n,100}]
Module[{nn=60},Mod[#[[1]],#[[2]]]&/@Thread[{PolygonalNumber[ 5,Range[ nn]],Accumulate[ Range[nn]]}]] (* Harvey P. Dale, Nov 19 2022 *)

def A175631(n): return 1+(-1)^n if (n<4) else 9*binomial(n/3, 2)
[A175631(n) for n in (1..60)] # G. C. Greubel, Jan 30 2022

For n>=3, a(n) = A000096(n-2).
From Chai Wah Wu, Oct 12 2018: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
G.f.: x^2*(2 - 6*x + 8*x^2 - 3*x^3)/(1 - x)^3. (End)
E.g.f.: (x/2)*(2 + 3*x - (2 - x)*exp(x)). - G. C. Greubel, Jan 30 2022

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A132318 Triangle, read by rows, where T(n,k) = [x^(k*2^(n-1))] Product_{i=0..n-1} (1 + x^(2^i))^(2^(n-i-1)) for n>0 with T(0,0)=1.

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A132316 a(n) = [x^n] Product_{i=0..n} (1 + x^(2^i) )^(2^(n-i)).

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A175631 a(n) = (n-th pentagonal number) modulo (n-th triangular number).

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