A368370
AGM transform of triangular numbers.
Original entry on oeis.org
0, 4, 514, 113920, 44084375, 28195584256, 28201577788944, 42039291307622400, 89651019067859485125, 264184676314240000000000, 1044528435971290465713291136, 5403748103160416506028643844096, 35801791315095649217787108688094375
Offset: 1
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Table[Sum[k*(k+1)/2, {k, 1, n}]^n - n^n*Product[k*(k+1)/2, {k, 1, n}], {n, 1, 12}] (* Vaclav Kotesovec, Jan 24 2024 *)
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from math import comb, factorial
def A368370(n): return comb(n+2,3)**n-(n**n*factorial(n)**2*(n+1)>>n) # Chai Wah Wu, Jan 25 2024
A369395
AGM transform of the even positive numbers.
Original entry on oeis.org
0, 4, 432, 61696, 12300000, 3339123264, 1195810789376, 549031054934016, 315439869711260160, 222215334010000000000, 188664842745174745939968, 190234762349632291168321536, 224946256003775354246877765632, 308520390288000443379128425267200, 486093585063330330624000000000000000
Offset: 1
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A369395[n_] := n^n*((n+1)^n - (2*n)!!);
Array[A369395, 15] (* Paolo Xausa, Jan 29 2024 *)
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a369395(n) = {my(v=vector(n,i,i+i)); vecsum(v)^n - n^n*vecprod(v)};
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from math import factorial
def A369395(n): return n**n*((n+1)**n-(factorial(n)<Chai Wah Wu, Jan 25 2024
A369698
AGM transform of positive cubes.
Original entry on oeis.org
0, 49, 40824, 96461056, 571250390625, 7338413252698641, 181953686508203782144, 7957561391610438862503936, 572547082070592542500791107625, 64157961305703333114506988525390625, 10714350425499230222239742740718898118656, 2571996060859292513876561308464753498396819456
Offset: 1
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A369698[n_] := (n*(n+1)/2)^(2*n) - n^n*n!^3; Array[A369698, 15]
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from math import factorial
def A369698(n): return ((n*(n+1))**(m:=n<<1)>>m) - n**n*factorial(n)**3 # Chai Wah Wu, Jan 29 2024
A369701
AGM transform of the numbers of partitions (A000041) of the positive numbers.
Original entry on oeis.org
0, 1, 54, 6961, 1233318, 487047961, 290742044714, 347251334512896, 683674076661539256, 2495297738110474036224, 14634026423059969492022144, 156866160296614402006202168641, 2612384850652790986902453089127552, 74149419491027435547521058057290511849, 3283295561194682488327071117265547706288707
Offset: 1
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A369701[n_] := With[{p = PartitionsP[Range[n]]}, Total[p]^n - n^n*Apply[Times, p]];
Array[A369701, 15]
A370223
AGM transform of the positive Fibonacci numbers.
Original entry on oeis.org
0, 0, 10, 865, 155082, 52802560, 40048988817, 71202718146816, 315615332953930528, 3574469013941010577249, 104798469697184132865547168, 7984603919946049180938300030976, 1584983603576817306123611193840098529, 820874582413458038007335015822715588591616
Offset: 1
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A370223[n_] := (Fibonacci[n+2]-1)^n - n^n*Fibonorial[n]; Array[A370223, 15]
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from itertools import count, islice
def A370223_gen(): # generator of terms
a, b, s, p = 1, 1, 0, 1
for n in count(1):
s += a
p *= a
yield s**n-n**n*p
a, b = b, a+b
A370223_list = list(islice(A370223_gen(),10)) # Chai Wah Wu, Feb 16 2024
A369393
a(n) = 2^n*(((n + 1)/2)^n - n!).
Original entry on oeis.org
0, 0, 1, 16, 241, 3936, 71569, 1452032, 32724801, 814205440, 22221533401, 661258764288, 21336094568881, 742703018860544, 27764596902369825, 1110071630916222976, 47289995917566900481, 2139290897163297619968, 102449006445196880700841, 5179102933596854288384000, 275667346790825720172556401
Offset: 0
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