This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The array starts in rows n>=2 and columns k>=1 as 1 5 15 35 70 126 210 330 495 715 1001 1365 1820 1 6 21 56 126 252 462 792 1287 2002 3003 4368 6188 1 7 27 77 182 378 714 1254 2079 3289 5005 7371 10556 1 8 33 98 238 504 966 1716 2871 4576 7007 10374 14924 1 9 39 119 294 630 1218 2178 3663 5863 9009 13377 19292 1 10 45 140 350 756 1470 2640 4455 7150 11011 16380 23660 1 11 51 161 406 882 1722 3102 5247 8437 13013 19383 28028 1 12 57 182 462 1008 1974 3564 6039 9724 15015 22386 32396 1 13 63 203 518 1134 2226 4026 6831 11011 17017 25389 36764
Table[Binomial[k + 3, 4] + (# - 2)*Binomial[k + 3, 5] &[m - k + 1], {m, 2, 12}, {k, m - 1}] // Flatten (* Michael De Vlieger, Nov 05 2020 *)
a(3) = 30 because 30 is the least integer which is pyramidal in 3 ways (30 is the fourth square pyramidal number, the third octagonal pyramidal number and also the second 31-gonal pyramidal number).
a(18) = 5 since 18 is a pentagonal pyramidal number, but not a square pyramidal or tetrahedral number.
a(10) = 2 because 10 is the third tetrahedral (or triangular pyramidal) number and also the second 9-gonal pyramidal number. a(30) = 3 because 30 is the fourth square pyramidal number, the third octagonal pyramidal number and also the second 29-gonal pyramidal number.
For 2 <= n <= 9, the n-gonal pyramidal number a(n) can be written as a product of smaller n-gonal pyramidal numbers in the following ways: n | a(n) --+------------------------------------- 2 | 36 = 6*6 3 | 560 = 4*4*35 = 10*56 4 | 4900 = 5*5*14*14 5 | 56448 = 196*288 6 | 4750 = 50*95 7 | 58372180608 = 196*456*653108 8 | 1130220 = 9*70*1794 9 | 6252757280000 = 10*10*10*80*78159466
For n = 5 we have: ---------------------------- 0 1 2 3 4 [5] ---------------------------- 0, 1, 5, 12, 22, 35, ... A000326 (pentagonal numbers) 0, 1, 6, 18, 40, 75, ... A002411 (pentagonal pyramidal numbers) 0, 1, 7, 25, 65, 140, ... A001296 (4-dimensional pyramidal numbers) 0, 1, 8, 33, 98, 238, ... A051836 (partial sums of A001296) 0, 1, 9, 42, 140, 378, ... A051923 (partial sums of A051836) 0, 1, 10, 52, 192, [570], ... A050494 (partial sums of A051923) ---------------------------- therefore a(5) = 570.
Table[n (n^2 - 2 n + 4) Binomial[2 n, n]/((n + 1) (n + 2)), {n, 0, 27}]
nmax = 27; CoefficientList[Series[(-4 + 31 x - 66 x^2 + 28 x^3 + (4 - 7 x) (1 - 4 x)^(3/2))/(2 x^2 (1 - 4 x)^(3/2)), {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[2 x] (4 - x + 2 x^2) BesselI[1, 2 x]/x - 2 Exp[2 x] (2 - x) BesselI[0, 2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[SeriesCoefficient[x (1 - 3 x + n x)/(1 - x)^(n + 3), {x, 0, n}], {n, 0, 27}]
A308488_vec(lim,J=10^6)={my( pyramid(s,n)=(3*n^2 + n^3*(s-2)-n*(s-5))/6, check(s)=j=if(lift(Mod(s,3))==2,((s-2)^2)/3-2,J);m=3;while(m<=j,if(ispolygonal(pyramid(s,m),s),return(pyramid(s,m)),m++));0); vector(lim,s,check(s+2))}
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