cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A274387 A diagonal of rectangular array A274391 of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x).

Original entry on oeis.org

1, 1, 3, 43, 1345, 71721, 5787931, 656778529, 99609347825, 19451450431009, 4752356577301171, 1419957082098657081, 509327639955159790777, 215968308944943346029577, 106859555896120941092549371, 61015970334444558798467062801, 39820542372512292977427634794721, 29454908124155520098406206592241281, 24512125500202005940687498958550124771, 22799363145943007981544986753209784020249, 23563018240183207044471748499194925348436201
Offset: 0

Views

Author

Paul D. Hanna, Jun 24 2016

Keywords

Comments

a(0) = 1 by convention. All terms appear to be odd.

Links

Crossrefs

Cf. A274391, A274388.

Programs

  • PARI
    {ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{A274391(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(exp(ITERATE(TREE,n,k)),k)}
/* Print table A274391 */
for(n=0,10,for(k=0,10,print1(A274391(n,k),", "));print("..."))
/* Print this sequence as a diagonal in A274391 */
for(n=0,20,print1(A274391(n-1,n),", "))

Formula

a(n) ~ c * (n-1)! * n! * exp(n), where c = 0.172... . - Vaclav Kotesovec, Jun 27 2016

A274388 Main diagonal of rectangular array A274391 of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x).

Original entry on oeis.org

1, 1, 5, 82, 2729, 151376, 12567187, 1457297878, 224869459201, 44538286061152, 11011493721321251, 3323602336722922574, 1202633627172086804257, 513869583003728865617848, 255985770924976071728925555, 147050140379016992236158750526, 96489590122440823908683879560193, 71722476615114676804476795900453248, 59952692198711311645811325484552353091, 55990325778560798795385664699772933184190, 58081532846176563089250398770056580653829601
Offset: 0

Views

Author

Paul D. Hanna, Jun 24 2016

Keywords

Links

Crossrefs

Cf. A274391, A274387, A274389.

Programs

  • PARI
    {ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{A274391(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(exp(ITERATE(TREE,n,k)),k)}
/* Print table A274391 */
for(n=0,10,for(k=0,10,print1(A274391(n,k),", "));print("..."))
/* Print this sequence as the main diagonal in A274391 */
for(n=0,20,print1(A274391(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

Views

Author

Paul D. Hanna, Jun 19 2016

Keywords

Comments

See table A274391 for the coefficients in exp( T^n(x) ), n>=0, where T^n(x) is the e.g.f. of the n-th row of this table.

Examples

			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) ); ...

Links

Crossrefs

Cf. A274391, A000169, A207833, A227278; diagonals: A274389, A274392.
Cf. A274570 (transforms diagonals).
Cf. A274740 (same table, but read differently).

Programs

  • PARI
    {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),", "));)

Formula

Let T^n(x) denote the n-th iteration of Euler's tree function T(x), then the coefficients in T^n(x) form the n-th row of this table, and the functions satisfy:
(1) T^n(x) = x * exp( Sum_{i=1..n} T^i(x) ).
(2) T^n(x) = T^(n-1)(x) * exp( T^n(x) ).
(3) T^n(x) = T^(n+1)( x/exp(x) ).

A274570 Triangle, read by rows, that transforms diagonals in the array A274390 of coefficients in successive iterations of Euler's tree function (A000169).

Original entry on oeis.org

1, 1, 1, 7, 2, 1, 127, 20, 3, 1, 4377, 470, 39, 4, 1, 245481, 19912, 1125, 64, 5, 1, 20391523, 1326382, 56505, 2188, 95, 6, 1, 2354116899, 127677580, 4354923, 127056, 3755, 132, 7, 1, 360734454993, 16767030632, 476265591, 11117244, 247465, 5922, 175, 8, 1, 70865037282673, 2880746218304, 70056231213, 1360983976, 24228925, 436632, 8785, 224, 9, 1, 17367953099244051, 627213971899610, 13329387478113, 221585119536, 3281909155, 47290506, 716457, 12440, 279, 10, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 28 2016

Keywords

Comments

This triangle also transforms diagonals in the array A274391 into each other, if we omit column 0 from those diagonals. The e.g.f. of row n of array A274391 equals exp(T^n(x)), where T^n(x) denotes the n-th iteration of Euler's tree function (A000169).

Examples

			This triangle T(n,k), n>=0, k=0..n, begins:
  1;
  1, 1;
  7, 2, 1;
  127, 20, 3, 1;
  4377, 470, 39, 4, 1;
  245481, 19912, 1125, 64, 5, 1;
  20391523, 1326382, 56505, 2188, 95, 6, 1;
  2354116899, 127677580, 4354923, 127056, 3755, 132, 7, 1;
  360734454993, 16767030632, 476265591, 11117244, 247465, 5922, 175, 8, 1;
  70865037282673, 2880746218304, 70056231213, 1360983976, 24228925, 436632, 8785, 224, 9, 1;
  17367953099244051, 627213971899610, 13329387478113, 221585119536, 3281909155, 47290506, 716457, 12440, 279, 10, 1;
  ...
Let D denote the triangular matrix defined by D(n,k) = T(n,k)/(n-k)!, such that D begins:
  1;
  1, 1;
  7/2!, 2, 1;
  127/3!, 20/2!, 3, 1;
  4377/4!, 470/3!, 39/2!, 4, 1;
  245481/5!, 19912/4!, 1125/3!, 64/2!, 5, 1;
  20391523/6!, 1326382/5!, 56505/4!, 2188/3!, 95/2!, 6, 1;
  ...
then D transforms diagonals in the array A274390 into each other:
  D * [1, 2/2, 30/3!, 948/4!, 50680/5!, 4090980/6!, ...] =
  [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...];
  D * [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...] =
  [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...];
  D * [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...] =
  [1, 8/2!, 165/3!, 6324/4!, 387205/5!, 34617288/6!, ...];
  ...
where array A274390 consists of coefficients in the iterations of Euler's tree function (A000169), and begins:
  1,  0,   0,     0,       0,        0,          0, ...;
  1,  2,   9,    64,     625,     7776,     117649, ...;
  1,  4,  30,   332,    4880,    89742,    1986124, ...;
  1,  6,  63,   948,   18645,   454158,   13221075, ...;
  1,  8, 108,  2056,   50680,  1537524,   55494712, ...;
  1, 10, 165,  3800,  112625,  4090980,  176238685, ...;
  1, 12, 234,  6324,  219000,  9266706,  463975764, ...;
  1, 14, 315,  9772,  387205, 18704322, 1067280319, ...;
  1, 16, 408, 14288,  637520, 34617288, 2217367600, ...;
  ...
Note that this triangle also transforms the diagonals of table A274391 into each other, if we omit column 0 from those diagonals.
After truncating column 0, table A274391 begins:
  1,  1,   1,     1,       1,         1,          1, ...;
  1,  3,  16,   125,    1296,     16807,     262144, ...;
  1,  5,  43,   525,    8321,    162463,    3774513, ...;
  1,  7,  82,  1345,   28396,    734149,   22485898, ...;
  1,  9, 133,  2729,   71721,   2300485,   87194689, ...;
  1, 11, 196,  4821,  151376,   5787931,  261066156, ...;
  1, 13, 271,  7765,  283321,  12567187,  656778529, ...;
  1, 15, 358, 11705,  486396,  24539593, 1457297878, ...;
  ...
for which the e.g.f. of row n equals exp(T^n(x)) - 1, where T^n(x) denotes the n-th iteration of Euler's tree function (A000169).
For example:
  D * [1, 3/2!, 43/3!, 1345/4!, 71721/5!, 5787931/6!, ...] =
  [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...];
  D * [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...] =
  [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ...];
  D * [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ...] =
  [1, 9/2!, 196/3!, 7765/4!, 486396/5!, 44223529/6!, ...];
  ...
The matrix inverse of triangle D, as shown with elements [D^-1][n,k] * (n-k)!, begins:
  1;
  -1, 1;
  -3, -2, 1;
  -40, -8, -3, 1;
  -1155, -140, -15, -4, 1;
  -57696, -5040, -324, -24, -5, 1;
  -4417175, -302092, -13923, -616, -35, -6, 1;
  -479964528, -26990720, -970848, -30720, -1040, -48, -7, 1;
  -70186001319, -3352727646, -98952435, -2439864, -58995, -1620, -63, -8, 1;
  -13284014648320, -551688200000, -13810202640, -279099200, -5254000, -102960, -2380, -80, -9, 1;
  -3158467118697099, -116039984093000, -2522473482375, -43202840076, -666167975, -10157796, -167475, -3344, -99, -10, 1;
  ...
The matrix square of triangle D, as shown with elements [D^2][n,k] * (n-k)!, begins:
  1;
  2, 1;
  18, 4, 1;
  377, 52, 6, 1;
  14304, 1414, 102, 8, 1;
  859977, 65904, 3411, 168, 10, 1;
  75306424, 4699274, 188496, 6668, 250, 12, 1;
  9061819643, 476161840, 15542811, 426144, 11485, 348, 14, 1;
  1435831150784, 65093379838, 1788015528, 39885108, 833280, 18162, 462, 16, 1;
  289948340816657, 11551390491440, 273593165397, 5134299808, 87266525, 1474704, 26999, 592, 18, 1;
  ...

Links

Crossrefs

Cf. A274390, A274571, A274572, A274573, A274574.

Programs

  • PARI
    {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 this triangle: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A274741 Table of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x), as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 16, 5, 1, 1, 1, 125, 43, 7, 1, 1, 1, 1296, 525, 82, 9, 1, 1, 1, 16807, 8321, 1345, 133, 11, 1, 1, 1, 262144, 162463, 28396, 2729, 196, 13, 1, 1, 1, 4782969, 3774513, 734149, 71721, 4821, 271, 15, 1, 1, 1, 100000000, 101808185, 22485898, 2300485, 151376, 7765, 358, 17, 1, 1, 1, 2357947691, 3129525793, 796769201, 87194689, 5787931, 283321, 11705, 457, 19, 1, 1, 1, 61917364224, 108063152091, 32084546824, 3815719969, 261066156, 12567187, 486396, 16785, 568, 21, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 04 2016

Keywords

Comments

See examples and formulas at A274391, which is the main entry for this table.
This entry is the same as table A274391, but read by antidiagonals from top down.

Examples

			See examples at A274391, which is the main entry for this table.
This table begins:
1, 1,  1,   1,     1,       1,         1,          1, ...;
1, 1,  3,  16,   125,    1296,     16807,     262144, ...;
1, 1,  5,  43,   525,    8321,    162463,    3774513, ...;
1, 1,  7,  82,  1345,   28396,    734149,   22485898, ...;
1, 1,  9, 133,  2729,   71721,   2300485,   87194689, ...;
1, 1, 11, 196,  4821,  151376,   5787931,  261066156, ...;
1, 1, 13, 271,  7765,  283321,  12567187,  656778529, ...;
1, 1, 15, 358, 11705,  486396,  24539593, 1457297878, ...;
...
This table may also be written as a triangle:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 16, 5, 1, 1;
1, 125, 43, 7, 1, 1;
1, 1296, 525, 82, 9, 1, 1;
1, 16807, 8321, 1345, 133, 11, 1, 1;
1, 262144, 162463, 28396, 2729, 196, 13, 1, 1;
1, 4782969, 3774513, 734149, 71721, 4821, 271, 15, 1, 1;
1, 100000000, 101808185, 22485898, 2300485, 151376, 7765, 358, 17, 1, 1;
...

Crossrefs

Cf. A274391.

Programs

  • PARI
    {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(exp(ITERATE(TREE, n, k)), k)}
/* Print this table as a rectangular array */
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
/* Print this table as a triangle */
for(n=0, 12, for(k=0, n, print1(T(k, n-k), ", "));print("") )
/* Print this table as a flattened array */
for(n=0, 12, for(k=0, n, print1(T(k, n-k), ", ")); )

Formula

See formulas at A274391, which is the main entry for this table.
Showing 1-5 of 5 results.

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