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 1-6 of 6 results.

A233530 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of the g.f. (A233531) such that column 0 consists of all zeros after row 1.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 8, 9, 4, 1, 0, 38, 40, 18, 5, 1, 0, 268, 264, 112, 30, 6, 1, 0, 2578, 2379, 953, 240, 45, 7, 1, 0, 31672, 27568, 10500, 2505, 440, 63, 8, 1, 0, 475120, 392895, 143308, 32686, 5445, 728, 84, 9, 1, 0, 8427696, 6663624, 2342284, 514660, 82176, 10423, 1120, 108, 10, 1, 0, 172607454, 131211423, 44677494, 9514570, 1467837, 178689, 18214, 1632, 135, 11, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 11 2013

Keywords

Examples

			Triangle begins:
1;
1, 1;
0, 2, 1;
0, 3, 3, 1;
0, 8, 9, 4, 1;
0, 38, 40, 18, 5, 1;
0, 268, 264, 112, 30, 6, 1;
0, 2578, 2379, 953, 240, 45, 7, 1;
0, 31672, 27568, 10500, 2505, 440, 63, 8, 1;
0, 475120, 392895, 143308, 32686, 5445, 728, 84, 9, 1;
0, 8427696, 6663624, 2342284, 514660, 82176, 10423, 1120, 108, 10, 1;
0, 172607454, 131211423, 44677494, 9514570, 1467837, 178689, 18214, 1632, 135, 11, 1;
0, 4008441848, 2943137604, 974898636, 202185010, 30319020, 3572037, 349720, 29718, 2280, 165, 12, 1; ...
in which column 0 consists of all zeros after row 1.
ILLUSTRATION OF GENERATING METHOD.
The g.f. of A233531 begins:
G(x) = x + x^2 - 2*x^3 + 6*x^4 - 18*x^5 + 44*x^6 - 56*x^7 - 300*x^8 + 2024*x^9 - 22022*x^10 - 130456*x^11 - 4241064*x^12 - 103538532*x^13 - 2893308780*x^14 - 88314189664*x^15 - 2924814872208*x^16 - 104538530634844*x^17 - 4010605941377292*x^18 +...
If we form a table of coefficients in the iterations of G(x) like so:
[1,  0,   0,   0,    0,     0,      0,      0,       0,        0, ...];
[1,  1,  -2,   6,  -18,    44,    -56,   -300,    2024,   -22022, ...];
[1,  2,  -2,   3,    2,   -48,    228,   -734,   -1298,   -14630, ...];
[1,  3,   0,  -3,   18,   -54,    -24,    625,   -6324,   -46064, ...];
[1,  4,   4,  -6,   12,    26,   -332,    244,   -2078,  -108754, ...];
[1,  5,  10,   0,  -10,    90,   -192,  -2044,   -3190,  -137176, ...];
[1,  6,  18,  21,  -18,    54,    312,  -3178,  -22032,  -203692, ...];
[1,  7,  28,  63,   42,   -28,    616,   -931,  -46722,  -457746, ...];
[1,  8,  40, 132,  248,   156,    504,   3144,  -51348,  -913356, ...];
[1,  9,  54, 234,  702,  1296,   1656,   6924,  -24444, -1366530, ...];
[1, 10,  70, 375, 1530,  4580,   9916,  22122,   38570, -1538042, ...];
[1, 11,  88, 561, 2882, 11814,  38280, 104929,  273592,  -987932, ...];
[1, 12, 108, 798, 4932, 25542, 110604, 407932, 1351614,  2563858, ...]; ...
then this triangle T transforms one diagonal in the above table into another:
T*[1, 1, -2, -3, 12, 90, 312, -931, -51348, -1366530, ...]
= [1, 2, 0, -6, -10, 54, 616, 3144, -24444, -1538042, ...];
T*[1, 2, 0, -6, -10, 54, 616, 3144, -24444, -1538042, ...]
= [1, 3, 4,  0, -18,-28, 504, 6924,  38570,  -987932, ...];
T*[1, 3, 4,  0, -18,-28, 504, 6924,  38570,  -987932, ...]
= [1, 4,10, 21,  42,156,1656,22122, 273592,  2563858, ...].

Crossrefs

Cf. A233531, A233532, A233533, A233534, A233535 (row sums).

Programs

  • PARI
    /* Given Root Series G, Calculate T(n,k) of Triangle: */
{T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x;
for(i=1, r+c-2, F=subst(F, x, G +x*O(x^(m+2)))); 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]); (P~*N~^-1)[n+1, k+1]}
/* Calculates Root Series G and then Prints ROWS of Triangle: */
{ROWS=12;V=[1,1];print("");print1("Root Sequence: [1, 1, ");
for(i=2,ROWS,V=concat(V,0);G=x*truncate(Ser(V));
for(n=0,#V-1,if(n==#V-1,V[#V]=-T(n,0));for(k=0,n, T(n,k)));print1(V[#V]", "););
print1("...]");print("");print("");print("Triangle begins:");
for(n=0,#V-2,for(k=0,n,print1(T(n,k),", "));print(""))}

A236961 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of g.f. of A236960 such that column 0 equals T(n,0) = n^n.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 27, 11, 3, 1, 256, 94, 21, 4, 1, 3125, 1076, 217, 34, 5, 1, 46656, 15362, 2910, 412, 50, 6, 1, 823543, 262171, 47598, 6333, 695, 69, 7, 1, 16777216, 5198778, 915221, 116768, 12045, 1082, 91, 8, 1, 387420489, 117368024, 20182962, 2498414, 247151, 20871, 1589, 116, 9, 1, 10000000000
Offset: 0

Views

Author

Paul D. Hanna, Feb 01 2014

Keywords

Examples

			This triangle begins:
1;
1, 1;
4, 2, 1;
27, 11, 3, 1;
256, 94, 21, 4, 1;
3125, 1076, 217, 34, 5, 1;
46656, 15362, 2910, 412, 50, 6, 1;
823543, 262171, 47598, 6333, 695, 69, 7, 1;
16777216, 5198778, 915221, 116768, 12045, 1082, 91, 8, 1;
387420489, 117368024, 20182962, 2498414, 247151, 20871, 1589, 116, 9, 1;
10000000000, 2970653234, 501463686, 60678776, 5824330, 471666, 33761, 2232, 144, 10, 1; ...
in which column 0 equals T(n,0) = n^n.
ILLUSTRATION.
This triangle transforms diagonals in the table of coefficients in the iterations of G(x), the g.f. of A236960, that starts as:
G(x) = x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 79*x^6 + 720*x^7 + 10735*x^8 + 211802*x^9 + 4968491*x^10 + 132655760*x^11 + 3943593218*x^12 +...
The table of coefficients in the successive iterations of G(x) begins:
[1,  0,   0,    0,     0,      0,       0,        0,         0, ...];
[1,  1,   2,    5,    16,     79,     720,    10735,    211802, ...];
[1,  2,   6,   21,    84,    410,    2876,    33235,    581074, ...];
[1,  3,  12,   54,   266,   1463,    9740,    90999,   1308954, ...];
[1,  4,  20,  110,   648,   4102,   28932,   248808,   2972926, ...];
[1,  5,  30,  195,  1340,   9705,   75264,   655599,   7059436, ...];
[1,  6,  42,  315,  2476,  20284,  174304,  1610487,  16952240, ...];
[1,  7,  56,  476,  4214,  38605,  366660,  3656975,  39586868, ...];
[1,  8,  72,  684,  6736,  68308,  712984,  7710392,  88021908, ...];
[1,  9,  90,  945, 10248, 114027, 1299696, 15223599, 185218134, ...];
[1, 10, 110, 1265, 14980, 181510, 2245428, 28396003, 369356822, ...]; ...
Then this triangle T transforms the adjacent diagonals in the above table into each other, as illustrated by:
T*[1, 1,  6,  54,  648,  9705, 174304, 3656975,  88021908, ...]
= [1, 2, 12, 110, 1340, 20284, 366660, 7710392, 185218134, ...];
T*[1, 2, 12, 110, 1340, 20284, 366660,  7710392, 185218134, ...]
= [1, 3, 20, 195, 2476, 38605, 712984, 15223599, 369356822, ...];
T*[1, 3, 20, 195, 2476, 38605,  712984, 15223599, 369356822, ...]
= [1, 4, 30, 315, 4214, 68308, 1299696, 28396003, 701068918, ...]; ...
RELATED TRIANGLE.
Compare this triangle to the triangle A088956(n,k) = (n-k+1)^(n-k-1)*C(n,k), that transforms diagonals in the table of coefficients in the iterations of x/(1-x):
1;
1, 1;
3, 2, 1;
16, 9, 3, 1;
125, 64, 18, 4, 1;
1296, 625, 160, 30, 5, 1;
16807, 7776, 1875, 320, 45, 6, 1; ...

Links

Crossrefs

Cf. A236960, A236962, A236963, A236964, A359716, A359717 (row sums).
Cf. variants: A233531, A088956.

Programs

  • PARI
    /* From Root Series G, Calculate T(n,k) of Triangle: */
{T(n, k) = my(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x;
for(i=1, r+c-2, F=subst(F, x, G +x*O(x^(m+2)))); 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]); (P~*N~^-1)[n+1, k+1]}
/* Calculates Root Series G and then Prints ROWS of Triangle: */
{ROWS=12;V=[1,1];print("");print1("Root Sequence: [1, 1, ");
for(i=2,ROWS,V=concat(V,0);G=x*truncate(Ser(V));
for(n=0,#V-1,if(n==#V-1,V[#V]=n^n-T(n,0));for(k=0,n, T(n,k)));print1(V[#V]", "););
print1("...]");print("");print("");print("Triangle begins:");
for(n=0,#V-2,for(k=0,n,print1(T(n,k),", "));print(""))}

A233533 Third column of triangle A233530.

Original entry on oeis.org

1, 3, 9, 40, 264, 2379, 27568, 392895, 6663624, 131211423, 2943137604, 74087543634, 2068127523264, 63395972293650, 2116776180704988, 76462961271015326, 2970720383283042732, 123517218432709179030, 5472063213488927240290, 257315864404023900340860, 12799687840583182179921204
Offset: 1

Views

Author

Paul D. Hanna, Dec 14 2013

Keywords

Crossrefs

Cf. A233530, A233531, A233532, A233534, A233535.

A233532 Second column of triangle A233530.

Original entry on oeis.org

1, 2, 3, 8, 38, 268, 2578, 31672, 475120, 8427696, 172607454, 4008441848, 104061288740, 2985810416824, 93812980749076, 3202899902480872, 118054380417848192, 4671653916923940952, 197528462484409468792, 8886812830338447522120, 423862241146246310158108, 21362480654281867734050272
Offset: 1

Views

Author

Paul D. Hanna, Dec 14 2013

Keywords

Crossrefs

Cf. A233530, A233531, A233533, A233534, A233535.

A233534 Fourth column of triangle A233530.

Original entry on oeis.org

1, 4, 18, 112, 953, 10500, 143308, 2342284, 44677494, 974898636, 23957348188, 654735747960, 19696461781336, 646719936551000, 23011823302544368, 881974963187446672, 36221318006594087010, 1586737195260479665112, 73850241923431162931872, 3638980616346240786818936
Offset: 1

Views

Author

Paul D. Hanna, Dec 14 2013

Keywords

Crossrefs

Cf. A233530, A233531, A233532, A233533, A233535.

A233535 Row sums of triangle A233530.

Original entry on oeis.org

1, 2, 3, 7, 22, 102, 681, 6203, 72757, 1050276, 18042102, 359677460, 8162936051, 207764278875, 5859801873585, 181362065062507, 6110186925475928, 222566852684792616, 8714776224549448369, 364990993734909935670, 16280201717192275756034, 770443051001734848973196, 38553687900969746513992234
Offset: 1

Views

Author

Paul D. Hanna, Dec 14 2013

Keywords

Crossrefs

Cf. A233530, A233531, A233532, A233533, A233534.
Showing 1-6 of 6 results.

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