A291846
Central terms in irregular triangle A291845.
Original entry on oeis.org
1, 1, 5, 33, 283, 2995, 37723, 551047, 9157923, 170606547, 3521075919, 79741123539, 1965955092517, 52414187219485, 1502559229282213, 46087421890091145, 1506033038595292467, 52232959640093489043, 1916263566511685329711, 74142047814365044902307, 3017192203838382727894241, 128829022389463544587355721, 5758871157244067281788041809
Offset: 0
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/* As Central Terms of Triangle A291845 */
{A291845(n, k)=polcoeff(prod(j=0, n-1, 1 + (2*j+1)*x + x^2), k)}
{for(n=0,25,print1(A291845(n,n),", "))}
A291847
A diagonal of irregular triangle A291845.
Original entry on oeis.org
1, 4, 26, 224, 2389, 30324, 446109, 7460928, 139775763, 2899264620, 65954625560, 1632654953280, 43688087178059, 1256602120453484, 38661480001233486, 1266934683224418816, 44054989554206606603, 1620147926716343851500, 62826169539072988352134, 2562071016044926371845920, 109611597248567432265872903, 4908887251696851858305862332
Offset: 0
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/* As a Diagonal in Triangle A291845 */
{A291845(n, k)=polcoeff(prod(j=0, n-1, 1 + (2*j+1)*x + x^2), k)}
{for(n=0,25,print1(A291845(n+1,n),", "))}
A291848
G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + (2*k+1)*x + x^2).
Original entry on oeis.org
1, 1, 2, 6, 15, 47, 150, 522, 1903, 7319, 29396, 122988, 534141, 2400061, 11136516, 53220492, 261576725, 1319629445, 6825232486, 36137198722, 195664517227, 1082169511883, 6108213101658, 35153836421302, 206126910439763, 1230477025952427, 7473067121404104, 46146114390128888, 289554642297817561, 1845220293901278041, 11936266843924805064
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 47*x^5 + 150*x^6 + 522*x^7 + 1903*x^8 + 7319*x^9 + 29396*x^10 + 122988*x^11 + 534141*x^12 +...
which equals the series:
A(x) = 1 + x*(1+x+x^2) + x^2*(1+x+x^2)*(1+3*x+x^2) + x^3*(1+x+x^2)*(1+3*x+x^2)*(1+5*x+x^2) + x^4*(1+x+x^2)*(1+3*x+x^2)*(1+5*x+x^2)*(1+7*x+x^2) +...
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nmax = 30; CoefficientList[Series[Sum[2^n*x^(2*n)*Pochhammer[(1 + x + x^2)/(2*x), n], {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2017 *)
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{a(n)=sum(k=0, n, polcoeff(prod(j=0, n-k-1, 1+(2*j+1)*x+x^2), k))}
for(n=0,30,print1(a(n),", "))
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{a(n)=polcoeff(sum(m=0, n, x^m*prod(j=0, m-1, 1+(2*j+1)*x+x^2))+x*O(x^n), n)}
for(n=0,25,print1(a(n),", "))
Showing 1-3 of 3 results.
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