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 1-3 of 3 results.

A253382 Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x-2k)^k.

Original entry on oeis.org

1, 5, 2, 5, 26, 3, 5, 170, 75, 4, 5, 810, 1035, 164, 5, 5, 3210, 10635, 3764, 305, 6, 5, 11274, 91275, 64244, 10385, 510, 7, 5, 36362, 693387, 910964, 261265, 24030, 791, 8, 5, 110090, 4822155, 11361908, 5422225, 830430, 49175, 1160, 9, 5, 317450, 31364235, 128935028, 98319505, 23510430, 2226455, 91880, 1629, 10
Offset: 0

Views

Author

Derek Orr, Dec 30 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x-0)^0 + T(n,1)*(x-2)^1 + T(n,2)*(x-4)^2 + ... + T(n,n)*(x-2n)^n, for n >= 0.

Examples

			From _Wolfdieter Lang_, Jan 14 2015: (Start)
The triangle T(n,k) starts:
n\k 0      1        2         3        4        5       6     7    8  9 ...
0:  1
1:  5
2:  5     26        3
3:  5    170       75         4
4:  5    810     1035       164        5
5:  5   3210    10635      3764      305        6
6:  5  11274    91275     64244    10385      510       7
7:  5  36362   693387    910964   261265    24030     791     8
8:  5 110090  4822155  11361908  5422225   830430   49175  1160    9
9:  5 317450 31364235 128935028 98319505 23510430 2226455 91880 1629 10
... Reformatted.
----------------------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 5*(x-0)^0 +  170*(x-2)^1 + 75*(x-4)^2 + 4*(x-6)^3. (End)

Crossrefs

Cf. A253381, A247236, A247237.

Programs

  • PARI
    T(n, k)=(k+1)-sum(i=k+1, n, (-2*i)^(i-k)*binomial(i, k)*T(n, i))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

Formula

T(n,n) = n+1, n >= 0.
T(n,n-1) = n + 2*n^2 + 2*n^3 = A046395(n), for n >= 1.
T(n,n-2) = (n-1)*(2*n^4-2*n^3-2*n^2-2*n+1), for n >= 2.
T(n,n-3) = (n-2)*(4*n^6-24*n^5+38*n^4-6*n^3+12*n^2-36*n+15)/3, for n >= 3.

Extensions

Edited. - Wolfdieter Lang, Jan 14 2015

A253383 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-3k)^k.

Original entry on oeis.org

1, 7, 2, 7, 38, 3, 7, 362, 111, 4, 7, 2522, 2271, 244, 5, 7, 14672, 34671, 8344, 455, 6, 7, 75908, 442911, 212464, 23135, 762, 7, 7, 361676, 5015199, 4498984, 869855, 53682, 1183, 8, 7, 1621388, 52044447, 83860840, 26997215, 2775282, 110047, 1736, 9, 7, 6935798, 505540767, 1423092160, 732435935, 117592782, 7458367, 205856, 2439, 10
Offset: 0

Views

Author

Derek Orr, Dec 30 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x-3)^0 + T(n,1)*(x-3)^1 + T(n,2)*(x-6)^2 + ... + T(n,n)*(x-3n)^n, for n >= 0.

Examples

			The triangle T(n,k) starts:
n\k 0        1         2         3         4        5       6     7  8  ...
0:  1
1:  7        2
2:  7       38         3
3:  7      362       111         4
4:  7     2522      2271       244         5
5:  7    14672     34671      8344       455        6
6:  7    75908    442911    212464     23135      762       7
7:  7   361676   5015199   4498984    869855    53682    1183     8
8:  7  1621388  52044447  83860840  26997215  2775282  110047  1736  9
...
-----------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 7*(x-0)^0 +  362*(x-3)^1 + 111*(x-6)^2 + 4*(x-9)^3.

Crossrefs

Cf. A253382, A247237.

Programs

  • PARI
    T(n, k)=(k+1)-sum(i=k+1, n, (-3*i)^(i-k)*binomial(i, k)*T(n, i))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

Formula

T(n,n-1) = n + 3*n^2 + 3*n^3, for n >= 1.
T(n,n-2) = (n-1)*(9*n^4 - 9*n^3 - 12*n^2 - 6*n + 2)/2, for n >= 2.
T(n,n-3) = (n-2)*(9*n^6 - 54*n^5 + 81*n^4 + 9*n^3 - 12*n^2 - 45*n + 14)/2, for n >= 3.

A253384 Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+3k)^k.

Original entry on oeis.org

1, -5, 2, -5, -34, 3, -5, 290, -105, 4, -5, -1870, 2055, -236, 5, -5, 10280, -30345, 7864, -445, 6, -5, -50956, 377895, -196256, 22235, -750, 7, -5, 234812, -4194393, 4090264, -824485, 52170, -1169, 8, -5, -1024900, 42834855, -75271592, 25302875, -2669430, 107695, -1720, 9
Offset: 0

Views

Author

Derek Orr, Dec 31 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^0 + T(n,1)*(x+3)^1 + T(n,2)*(x+6)^2 + ... + T(n,n)*(x+3n)^n, for n >= 0.

Examples

			The triangle T(n,k) starts:
n\k  0       1         2        3        4      5      6    7 ...
0:   1
1:  -5       2
2:  -5     -34         3
3:  -5     290      -105        4
4:  -5   -1870      2055     -236        5
5:  -5   10280    -30345     7864     -445      6
6:  -5  -50956    377895  -196256    22235   -750      7
7:  -5  234812  -4194393  4090264  -824485  52170  -1169   8
...
-----------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = -5*(x+0)^0 +  290*(x+3)^1 - 105*(x+6)^2 + 4*(x+9)^3.

Crossrefs

Cf. A247236, A247237, A253381, A253382.

Programs

  • PARI
    T(n, k)=(k+1)-sum(i=k+1, n, (3*i)^(i-k)*binomial(i, k)*T(n, i))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

Formula

T(n,n) = n + 1, n >= 0.
T(n,n-1) = n - 3*n^2 - 3*n^3, for n >= 1.
T(n,n-2) = (n-1)*(9*n^4 - 9*n^3 - 24*n^2 + 6*n + 2)/2, for n >= 2.
T(n,n-3) = (2-n)*(9*n^6 - 54*n^5 + 63*n^4 + 99*n^3 - 138*n^2 + 9*n + 10)/2, for n >= 3.

Extensions

Edited; name changed, cross references added. - Wolfdieter Lang, Jan 22 2015
Showing 1-3 of 3 results.

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