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 1-4 of 4 results.

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(""))

A274572 Column 1 of triangle A274570.

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

Views

Author

Paul D. Hanna, Jun 28 2016

Keywords

Comments

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

Links

Crossrefs

Cf. A274570, A274571, 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 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),", "))

A274573 Column 2 of triangle A274570.

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

Views

Author

Paul D. Hanna, Jun 28 2016

Keywords

Comments

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

Links

Crossrefs

Cf. A274570, A274571, A274572, 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 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),", "))

A274574 Central terms of triangle A274570.

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

Views

Author

Paul D. Hanna, Jun 28 2016

Keywords

Comments

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

Crossrefs

Cf. A274570, A274571, A274572, A274573.

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 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),", "))
Showing 1-4 of 4 results.

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