A107045
Numerators of the triangle of coefficients T(n,k), read by rows, that satisfy: y^x = Sum_{n=0..x} R_n(y)*x^n for all nonnegative integers x, y, where R_n(y) = Sum_{k=0..n} T(n,k)*y^k and T(n,k) = a(n,k)/A107046(n,k).
Original entry on oeis.org
1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -11, 1, 1, -1, 1, -677, -61, 7, 1, -1, 1, 15311, -259, -1, 7, 1, -1, 1, 1170049273, -971891, -54407, 407, 23, 1, -1, 1, 541293087149, 426148171, -15993079, -58573, 829, 17, 1, -1, 1, -15074636799365429, 31108643619709, -23328513449, -138374321, -53429, 1501, 47, 1
Offset: 0
These are the numerators of the triangle that begins:
1;
-1,1;
1/4,-1/2,1/4;
-1/108,1/18,-1/12,1/27;
-11/6912,1/576,1/192,-1/108,1/256;
-677/21600000,-61/360000,7/24000,1/2700,-1/1280,1/3125; ...
which equals the matrix inverse of triangle A079901(n,k)=n^k:
1;
1,1;
1,2,4;
1,3,9,27;
1,4,16,64,256;
1,5,25,125,625,3125; ...
-
a(n,k)=numerator((matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1])
A107046
Denominators of the triangle of coefficients T(n,k), read by rows, that satisfy: y^x = Sum_{n=0..x} R_n(y)*x^n for all nonnegative integers x, y, where R_n(y) = Sum_{k=0..n} T(n,k)*y^k and T(n,k) = A107045(n,k)/a(n,k).
Original entry on oeis.org
1, 1, 1, 4, 2, 4, 108, 18, 12, 27, 6912, 576, 192, 108, 256, 21600000, 360000, 24000, 2700, 1280, 3125, 2332800000, 12960000, 2592000, 291600, 46080, 18750, 46656, 1921161110400000, 1524731040000, 43563744000, 700131600, 15805440, 918750
Offset: 0
These are the denominators of the triangle that begins:
1;
-1,1;
1/4,-1/2,1/4;
-1/108,1/18,-1/12,1/27;
-11/6912,1/576,1/192,-1/108,1/256;
-677/21600000,-61/360000,7/24000,1/2700,-1/1280,1/3125; ...
which equals the matrix inverse of triangle A079901(n,k)=n^k:
1;
1,1;
1,2,4;
1,3,9,27;
1,4,16,64,256;
1,5,25,125,625,3125; ...
-
a(n,k)=denominator((matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1])
A107047
Numerators of coefficients that satisfy: 2^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = a(k)/A107048(k).
Original entry on oeis.org
1, 1, 1, 7, 77, 32387, 395159, 31824093937, 44855117331581, 1825389561156191099, 1571879809058619206897, 28070265610073576492663157851903, 2782861136717279135850604073374039
Offset: 0
2^0 = 1;
2^1 = 1 + 1;
2^2 = 1 + 1*2 + (1/4)*2^2;
2^3 = 1 + 1*3 + (1/4)*3^2 + (7/108)*3^3;
2^4 = 1 + 1*4 + (1/4)*4^2 + (7/108)*4^3 + (77/6912)*4^4.
Initial fractional coefficients are:
A107047/A107048 = {1, 1, 1/4, 7/108, 77/6912, 32387/21600000,
395159/2332800000, 31824093937/1921161110400000,
44855117331581/31476303632793600000, ... }.
-
{a(n)=numerator(sum(k=0,n,2^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
A107048
Denominators of coefficients that satisfy: 2^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = A107047(k)/a(k).
Original entry on oeis.org
1, 1, 4, 108, 6912, 21600000, 2332800000, 1921161110400000, 31476303632793600000, 16727798278915463577600000, 209097478486443294720000000000
Offset: 0
2^0 = 1;
2^1 = 1 + 1;
2^2 = 1 + 1*2 + (1/4)*2^2;
2^3 = 1 + 1*3 + (1/4)*3^2 + (7/108)*3^3;
2^4 = 1 + 1*4 + (1/4)*4^2 + (7/108)*4^3 + (77/6912)*4^4.
Initial fractional coefficients are:
A107047/A107048 = {1, 1, 1/4, 7/108, 77/6912, 32387/21600000,
395159/2332800000, 31824093937/1921161110400000,
44855117331581/31476303632793600000, ... }.
-
{a(n)=denominator(sum(k=0,n,2^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
A107049
Numerators of coefficients that satisfy: 3^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = a(k)/A107050(k).
Original entry on oeis.org
1, 2, 1, 11, 101, 71723, 1462111, 194269981673, 224103520039487, 14876670160046176873, 20871062802926443547323, 606768727432357137728440774281877, 97827345788163051844748893917483101
Offset: 0
3^0 = 1;
3^1 = 1 + (2)*1;
3^2 = 1 + (2)*2 + (1)*2^2;
3^3 = 1 + (2)*3 + (1)*3^2 + (11/27)*3^3;
3^4 = 1 + (2)*4 + (1)*4^2 + (11/27)*4^3 + (101/864)*4^4.
Initial coefficients are:
A107049/A107050 = {1, 2, 1, 11/27, 101/864, 71723/2700000,
1462111/291600000, 194269981673/240145138800000,
224103520039487/1967268977049600000, ...}.
-
{a(n)=numerator(sum(k=0,n,3^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
A107050
Denominators of coefficients that satisfy: 3^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = A107049(k)/a(k).
Original entry on oeis.org
1, 1, 1, 27, 864, 2700000, 291600000, 240145138800000, 1967268977049600000, 1045487392432216473600000, 13068592405402705920000000000, 3728621931719673008255139717120000000000
Offset: 0
3^0 = 1;
3^1 = 1 + (2)*1;
3^2 = 1 + (2)*2 + (1)*2^2;
3^3 = 1 + (2)*3 + (1)*3^2 + (11/27)*3^3;
3^4 = 1 + (2)*4 + (1)*4^2 + (11/27)*4^3 + (101/864)*4^4.
Initial coefficients are:
A107049/A107050 = {1, 2, 1, 11/27, 101/864, 71723/2700000,
1462111/291600000, 194269981673/240145138800000,
224103520039487/1967268977049600000, ...}.
-
{a(n)=denominator(sum(k=0,n,3^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
A107051
Numerators of coefficients that satisfy: 4^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = a(k)/A107052(k).
Original entry on oeis.org
1, 3, 9, 5, 127, 124273, 385829, 70009765747, 220026935042111, 59574747365570286907, 113453152114585319883313, 4471148647570383262775217527741887
Offset: 0
4^0 = 1;
4^1 = 1 + (3)*1;
4^2 = 1 + (3)*2 + (9/4)*2^2;
4^3 = 1 + (3)*3 + (9/4)*3^2 + (5/4)*3^3;
4^4 = 1 + (3)*4 + (9/4)*4^2 + (5/4)*4^3 + (127/256)*4^4.
Initial coefficients are:
A107051/A107052 = {1, 3, 9/4, 5/4, 127/256, 124273/800000,
385829/9600000, 70009765747/7906012800000,
220026935042111/129532113715200000, ...}.
-
{a(n)=numerator(sum(k=0,n,4^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
A107052
Denominators of coefficients that satisfy: 4^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = A107051(k)/a(k).
Original entry on oeis.org
1, 1, 4, 4, 256, 800000, 9600000, 7906012800000, 129532113715200000, 206516028134758809600000, 2581450351684485120000000000, 736517912438453927556570808320000000000
Offset: 0
4^0 = 1;
4^1 = 1 + (3)*1;
4^2 = 1 + (3)*2 + (9/4)*2^2;
4^3 = 1 + (3)*3 + (9/4)*3^2 + (5/4)*3^3;
4^4 = 1 + (3)*4 + (9/4)*4^2 + (5/4)*4^3 + (127/256)*4^4.
Initial coefficients are:
A107051/A107052 = {1, 3, 9/4, 5/4, 127/256, 124273/800000,
385829/9600000, 70009765747/7906012800000,
220026935042111/129532113715200000, ...}.
-
{a(n)=denominator(sum(k=0,n,4^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
A107053
Numerators of coefficients that satisfy: 5^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = a(k)/A107054(k).
Original entry on oeis.org
1, 4, 4, 76, 307, 380989, 13464073, 3084163593839, 6109976845914041, 694491088545589897439, 1664245369537759004769053, 82473629015170976645702130970352147
Offset: 0
5^0 = 1;
5^1 = 1 + (4)*1;
5^2 = 1 + (4)*2 + (4)*2^2;
5^3 = 1 + (4)*3 + (4)*3^2 + (76/27)*3^3;
5^4 = 1 + (4)*4 + (4)*4^2 + (76/27)*4^3 + (307/216)*4^4.
Initial coefficients are:
A107053/A107054 = {1, 4, 4, 76/27, 307/216, 380989/675000,
13464073/72900000, 3084163593839/60036284700000,
6109976845914041/491817244262400000, ...}
-
{a(n)=numerator(sum(k=0,n,5^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
A107054
Denominators of coefficients that satisfy: 5^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = A107053(k)/a(k).
Original entry on oeis.org
1, 1, 1, 27, 216, 675000, 72900000, 60036284700000, 491817244262400000, 261371848108054118400000, 3267148101350676480000000000, 932155482929918252063784929280000000000
Offset: 0
5^0 = 1;
5^1 = 1 + (4)*1;
5^2 = 1 + (4)*2 + (4)*2^2;
5^3 = 1 + (4)*3 + (4)*3^2 + (76/27)*3^3;
5^4 = 1 + (4)*4 + (4)*4^2 + (76/27)*4^3 + (307/216)*4^4.
Initial coefficients are:
A107053/A107054 = {1, 4, 4, 76/27, 307/216, 380989/675000,
13464073/72900000, 3084163593839/60036284700000,
6109976845914041/491817244262400000, ...}
-
{a(n)=denominator(sum(k=0,n,5^k*(matrix(n+1,n+1,r,c,if(r>=c,(r-1)^(c-1)))^-1)[n+1,k+1]))}
Showing 1-10 of 15 results.
Comments