A145085 Square table, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(n+1) with R(n,0) = 1; that is, the logarithmic derivative of the e.g.f. of row n equals the e.g.f. of row n+1 to the n+1 power, for n>=0.

1, 1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 17, 39, 1, 1, 5, 31, 151, 322, 1, 1, 6, 49, 373, 1901, 3723, 1, 1, 7, 71, 741, 6250, 31851, 57577, 1, 1, 8, 97, 1291, 15457, 136711, 680265, 1147188, 1, 1, 9, 127, 2059, 32186, 416661, 3740137, 17947631, 28557909, 1, 1, 10, 161, 3081, 59677, 1030491, 13908049, 124143598, 571101141, 866222535

Offset: 0

Author

Paul D. Hanna, Oct 01 2008

Keywords

• nonn
• tabl

Examples

			Table begins:
1,1,2,7,39,322,3723,57577,1147188,28557909,866222535,31362744620,...;
1,1,3,17,151,1901,31851,680265,17947631,571101141,21507723971,...;
1,1,4,31,373,6250,136711,3740137,124143598,4887140221,224203589593,...;
1,1,5,49,741,15457,416661,13908049,557865765,26296627233,...;
1,1,6,71,1291,32186,1030491,40606281,1911466016,105145651821,...;
1,1,7,97,2059,59677,2211823,100479577,5431432483,341787359269,...;
1,1,8,127,3081,101746,4283511,220384585,13453788426,953539677861,...;
1,1,9,161,4393,162785,7672041,440897697,30000376553,2365207145121,...;
1,1,10,199,6031,247762,12921931,820341289,61561430380,5344379824933,...;
1,1,11,241,8031,362221,20710131,1439328361,118089834231,11194348009941,...;
1,1,12,287,10429,512282,31860423,2405825577,214232473478,22019097106029,..;
1,1,13,337,13261,704641,47357821,3860734705,370824076621,41076472798081,..;
1,1,14,391,16563,946570,68362971,5983992457,616668950808,73237232298621,..;
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1080

Crossrefs

Cf. A145080; rows: A145086, A145087, A145088, A145089.

Programs

PARI

{T(n,k)=local(A=vector(n+k+2,j,1+j*x)); for(i=0,n+k+1,for(j=0,n+k,m=n+k+1-j;A[m]=exp(intformal(A[m+1]^m+x*O(x^k))))); k!*polcoeff(A[n+1],k,x)}
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

Formula

Row e.g.f.s satisfy: R(n,x) = exp( Integral R(n+1,x)^(n+1) dx ).

Row e.g.f.s satisfy: R(n,x) = 1 + Integral R(n,x)*R(n+1,x)^(n+1) dx.

Row e.g.f.s satisfy: R'(n,x)/R(n,x) = R(n+1,x)^(n+1) with R(n,0) = 1.

Extensions

Entry corrected by Paul D. Hanna, Sep 22 2020