Original entry on oeis.org
1, 1, 7, 127, 4377, 245481, 20391523, 2354116899, 360734454993, 70865037282673, 17367953099244051, 5195706189463681299, 1863485648739527246569, 789370246456579516316121, 389929989984983783920348611, 222178771969671609391720410691, 144648509476124539709343154760897, 106712830948451223242311469280356609, 88557950557114913966623605248882438755, 82132537612235618834557329353828430430755
Offset: 0
-
{T(n, k)=local(F=x,
LW=serreverse(x*exp(-x+x*O(x^(n+2)))), M, N, P, m=max(n, k));
M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, LW)); polcoeff(F, c));
N=matrix(m+1, m+1, r, c, M[r, c]);
P=matrix(m+1, m+1, r, c, M[r+1, c]);
(n-k)!*(P~*N~^-1)[n+1, k+1]}
/* Print triangle A274570: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is column 0 */
for(n=0,20,print1(T(n,0),", "))
Original entry on oeis.org
1, 2, 20, 470, 19912, 1326382, 127677580, 16767030632, 2880746218304, 627213971899610, 168767535712912684, 54994347890521005100, 21342142821229037730064, 9726400286221416303901358, 5143644030714149522751534524, 3124088412968372614077895431788, 2159818183532141245447039295746240, 1686295004858842334963772859214802354, 1476540037893212558044217633785452773068, 1440964034588041764141738802548853847618732
Offset: 0
-
{T(n, k)=local(F=x,
LW=serreverse(x*exp(-x+x*O(x^(n+2)))), M, N, P, m=max(n, k));
M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, LW)); polcoeff(F, c));
N=matrix(m+1, m+1, r, c, M[r, c]);
P=matrix(m+1, m+1, r, c, M[r+1, c]);
(n-k)!*(P~*N~^-1)[n+1, k+1]}
/* Print triangle A274570: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is column 1 */
for(n=0,20,print1(T(n+1,1),", "))
Original entry on oeis.org
1, 3, 39, 1125, 56505, 4354923, 476265591, 70056231213, 13329387478113, 3184105899176739, 932720103991595919, 328710383679927689157, 137188135970104212440721, 66909975066823379752742835, 37706189062081696231083478647, 24312515006613477431766856702797, 17784145956730483348850500758855969, 14647274671009402833580157237962722147, 13492886769176913829412675003231182928079
Offset: 0
-
{T(n, k)=local(F=x,
LW=serreverse(x*exp(-x+x*O(x^(n+2)))), M, N, P, m=max(n, k));
M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, LW)); polcoeff(F, c));
N=matrix(m+1, m+1, r, c, M[r, c]);
P=matrix(m+1, m+1, r, c, M[r+1, c]);
(n-k)!*(P~*N~^-1)[n+1, k+1]}
/* Print triangle A274570: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is column 2 */
for(n=0,20,print1(T(n+2,2),", "))
Original entry on oeis.org
1, 2, 39, 2188, 247465, 47290506, 13732594855, 5645761143968, 3124313624563281, 2240929551882269890, 2023001689428835457551, 2245340983227461222262600, 3005921392102922941037743561, 4777188534537036418050441999458, 8892651921874938986718539648539335, 19167346139929272962512547586833106016, 47363669252787891219004826832547428944065, 133017373943189884985366059167726505579520322, 421334607602498277189468756234637306051458074191, 1495034827615578030423476599123008111000477082402040, 5906697677063490360959940664316005473632429506949424681
Offset: 0
-
{T(n, k)=local(F=x,
LW=serreverse(x*exp(-x+x*O(x^(n+2)))), M, N, P, m=max(n, k));
M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, LW)); polcoeff(F, c));
N=matrix(m+1, m+1, r, c, M[r, c]);
P=matrix(m+1, m+1, r, c, M[r+1, c]);
(n-k)!*(P~*N~^-1)[n+1, k+1]}
/* Print triangle : */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is central terms */
for(n=0,20,print1(T(2*n,n),", "))
A274390
Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 4, 9, 0, 1, 6, 30, 64, 0, 1, 8, 63, 332, 625, 0, 1, 10, 108, 948, 4880, 7776, 0, 1, 12, 165, 2056, 18645, 89742, 117649, 0, 1, 14, 234, 3800, 50680, 454158, 1986124, 2097152, 0, 1, 16, 315, 6324, 112625, 1537524, 13221075, 51471800, 43046721, 0, 1, 18, 408, 9772, 219000, 4090980, 55494712, 448434136, 1530489744, 1000000000, 0, 1, 20, 513, 14288, 387205, 9266706, 176238685, 2325685632, 17386204761, 51395228090, 25937424601, 0, 1, 22, 630, 20016, 637520, 18704322, 463975764, 8793850560, 111107380464, 759123121050, 1924687118684, 743008370688, 0, 1, 24, 759, 27100, 993105, 34617288, 1067280319, 26858490392, 499217336145, 5964692819140, 36882981687519, 79553145323940, 23298085122481, 0
Offset: 0
This table begins:
1, 0, 0, 0, 0, 0, 0, 0, ...;
1, 2, 9, 64, 625, 7776, 117649, 2097152, ...;
1, 4, 30, 332, 4880, 89742, 1986124, 51471800, ...;
1, 6, 63, 948, 18645, 454158, 13221075, 448434136, ...;
1, 8, 108, 2056, 50680, 1537524, 55494712, 2325685632, ...;
1, 10, 165, 3800, 112625, 4090980, 176238685, 8793850560, ...;
1, 12, 234, 6324, 219000, 9266706, 463975764, 26858490392, ...;
1, 14, 315, 9772, 387205, 18704322, 1067280319, 70311813880, ...;
1, 16, 408, 14288, 637520, 34617288, 2217367600, 163802295616, ...;
1, 18, 513, 20016, 993105, 59879304, 4254311817, 348285415872, ...;
1, 20, 630, 27100, 1480000, 98110710, 7656893020, 688058734520, ...;
...
where the e.g.f.s of the rows are iterations of T(x) and begin:
T^0(x) = x;
T^1(x) = T(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...;
T^2(x) = T(T(x)) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! + 89742*x^6/6! + 1986124*x^7/7! + 51471800*x^8/8! +...+ A207833(n)*x^n/n! +...;
T^3(x) = T(T(T(x))) = x + 6*x^2/2! + 63*x^3/3! + 948*x^4/4! + 18645*x^5/5! + 454158*x^6/6! + 13221075*x^7/7! + 448434136*x^8/8! +...+ A227278(n)*x^n/n! +...;
T^4(x) = T(T(T(T(x)))) = x + 8*x^2/2! + 108*x^3/3! + 2056*x^4/4! + 50680*x^5/5! + 1537524*x^6/6! + 55494712*x^7/7! + 2325685632*x^8/8! +...;
...
where T^n(x)/exp( T^n(x) ) = T^n( x/exp(x) ) = T^(n-1)(x).
Also we have
T(x) = x*exp( T(x) );
T^2(x) = x*exp( T(x) + T^2(x) );
T^3(x) = x*exp( T(x) + T^2(x) + T^3(x) );
T^4(x) = x*exp( T(x) + T^2(x) + T^3(x) + T^4(x) ); ...
Cf.
A274570 (transforms diagonals).
Cf.
A274740 (same table, but read differently).
-
{ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{T(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(ITERATE(TREE,n,k),k)}
/* Print this table as a square array */
for(n=0,10,for(k=1,10,print1(T(n,k),", "));print(""))
/* Print this table as a flattened array */
for(n=0,12,for(k=1,n,print1(T(n-k,k),", "));)
Showing 1-5 of 5 results.
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