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.

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A213953 Triangle by rows, inverse of A208891.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, -2, 5, 0, -3, -1, 1, -9, 5, 10, -2, -4, -1, 1, -9, -21, 25, 15, -5, -5, -1, 1, 50, -105, -11, 62, 19, -9, -6, -1, 1, 267, -141, -301, 56, 119, 21, -14, -7, -1, 1, 413, 777
Offset: 0

Views

Author

Gary W. Adamson, Jun 26 2012

Keywords

Examples

			Triangle starts:
1;
-1, 1
0, -1, 1
1, -1, -1, 1;
1, 1, -2, -1, 1;
-2, 5, 0, -3, -1, 1;
-9, 5, 10, -2, -4, -1, 1;
-9, -21, 25, 15, -5, -5, -1, 1;
50, -105, -11, 62, 19, -9, -6, -1, 1;
267, -141, -301, 56, 119, 21, -14, -7, -1, 1;
413, 777, -1040, -566, 226, 198, 20, -20, -8, -1, 1;
...

Crossrefs

Cf. A208891, A000587 (first column), A014619 (2nd column), A080956 (4th subdiagonal).

Programs

  • Maple
    A208891 := proc(n,k)
    if n <0 or k<0 or k>n then
            0;
    elif n = k then
            1 ;
    else
            binomial(n-1,k) ;
    end if;
end proc:
A259456 := proc(n)
    local A, row, col ;
    A := Matrix(n, n) ;
    for row from 1 to n do
        for col from 1 to n do
            A[row, col] := A208891(row-1,col-1) ;
        end do:
    end do:
    LinearAlgebra[MatrixInverse](A) ;
end proc:
A259456(20) ; # R. J. Mathar, Jul 21 2015

Formula

Inverse of triangle A208891, Pascal's triangle matrix with an appended right border of 1's.

A341331 a(n) = n^n - (n-1)^n - (n-2)^n - ... - 1^n.

Original entry on oeis.org

1, 3, 18, 158, 1825, 26141, 446782, 8869820, 200535993, 5085658075, 142947350986, 4410243535402, 148156328308105, 5382924338773177, 210309307208574750, 8791961076113491704, 391581231268402937041, 18510377905675629883959, 925555262359725659407258
Offset: 1

Views

Author

Seiichi Manyama, Feb 09 2021

Keywords

Links

Crossrefs

Cf. A000312, A031971, A080956, A121706, A179297, A179298, A173142, A191686, A290844.

Programs

  • Maple
    f:= proc(n) local k;  n^n - add(k^n,k=1..n-1) end proc:
map(f, [$1..30]); # Robert Israel, Feb 10 2021
  • Mathematica
    a[n_] := n^n - Sum[k^n, {k, 0, n - 1}]; Array[a, 20] (* Amiram Eldar, Apr 28 2021 *)
  • PARI
    a(n) = n^n-sum(k=0, n-1, k^n);

Formula

a(n) = A000312(n) - A121706(n).
a(n) = - A290844(n-1,n) for n > 1.

A134565 Expansion of reversion of (x - 2*x^2) / (1 - x)^3.

Original entry on oeis.org

1, -1, 2, -3, 7, -12, 30, -55, 143, -273, 728, -1428, 3876, -7752, 21318, -43263, 120175, -246675, 690690, -1430715, 4032015, -8414640, 23841480, -50067108, 142498692, -300830572, 859515920, -1822766520, 5225264024, -11124755664, 31983672534, -68328754959
Offset: 1

Views

Author

Michael Somos, Nov 01 2007

Keywords

Examples

			G.f. = x - x^2 + 2*x^3 - 3*x^4 + 7*x^5 - 12*x^6 + 30*x^7 - 55*x^8 + 143*x^9 + ...

Links

Crossrefs

Cf. A001764, A005161, A006013, A047749, A080956, A099576.

Programs

  • Mathematica
    a[ n_] := With[ {m = Quotient[n, 2]}, If[n < 1, 0, -(-1)^n Binomial[n + m, n - m] / (2 m + 1)]]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := If[n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[(x - 2 x^2) / (1 - x)^3, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Oct 16 2015 *)
a[n_] := (-1)^(n-1)*Binomial[2*n, n-1]*Hypergeometric2F1[-n+1, n, -2*n, -1] / n; Flatten[Table[a[n], {n, 1, 32}]] (* Detlef Meya, Dec 26 2023 *)
  • PARI
    {a(n) = my( m = n\2); if( n<1, 0, -(-1)^n * binomial( n + m, n - m) / (2 * m + 1))};
  • PARI
    {a(n) = if( n<1, 0, polcoeff( serreverse( (x - 2 * x^2) / (1 - x)^3 + x * O(x^n) ), n))};
  • PARI
    {a(n) = if( n<1, 0, polcoeff( 1 / ( 1 + 1 / serreverse( x - x^3 + x * O(x^n) )), n))};

Formula

Given g.f. A(x), then 1 = (1/A(x) + 1/A(-x)) / 2.
a(n) = -(-1)^n * binomial(n + m, n - m) / (2*m + 1) where m = floor(n/2) if n>0.
From Michael Somos, Apr 13 2012 (Start)
a(n) = -(-1)^n * A047749(n) unless n=0. a(2*n) = - A001764(n) unless n=0. a(2*n + 1) = A006013(n).
Reversion of A080956 with offset 1.
Hankel transform is A005161 omitting first 1.
n * a(n) = -(-1)^n * A099576(n-1). (End)
D-finite with recurrence +8*n*(n+1)*a(n) -36*n*(n-2)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Sep 24 2021
a(n) = (-1)^(n-1)*binomial(2*n, n-1)*hypergeom([-n+1, n], [-2*n], -1) / n. - Detlef Meya, Dec 26 2023

A136261 Triangle T(n,k) = k*A122188(n,k), read by rows.

Original entry on oeis.org

-1, -1, 2, 1, 2, -3, -1, -2, -3, 4, 1, 2, 3, 4, -5, -1, -2, -3, -4, -5, 6, 1, 2, 3, 4, 5, 6, -7, -1, -2, -3, -4, -5, -6, -7, 8, 1, 2, 3, 4, 5, 6, 7, 8, -9, -1, -2, -3, -4, -5, -6, -7, -8, -9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, -11
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Mar 18 2008

Keywords

Comments

Multiplication of the columns of A122188 by their index is equivalent to differentiation of the polynomials B(n,x) defined in A122188.
Row sums are -1, 1, 0, -2, 5, -9, 14, -20, 27, -35, 44, ... =(-1)^n*A080956(n-1).

Examples

			  -1;
  -1,  2;
   1,  2, -3;
  -1, -2, -3,  4;
   1,  2,  3,  4, -5;
  -1, -2, -3, -4, -5,  6;
   1,  2,  3,  4,  5,  6, -7;
  -1, -2, -3, -4, -5, -6, -7,  8;
   1,  2,  3,  4,  5,  6,  7,  8, -9;
  -1, -2, -3, -4, -5, -6, -7, -8, -9, 10;
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, -11;

Crossrefs

Cf. A002260, A122188.

Programs

  • Mathematica
    Clear[B, x, n] B[x, 0] = 1; B[x, 1] = -x + 1; B[x_, n_] := B[x, n] = If[n > 1, (-1)^n*(x^n - Sum[x^m, {m, 0, n - 1}])]; P[x_, n_] := D[B[x, n + 1], x]; Table[ExpandAll[P[x, n]], {n, 0, 10}]; a = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 10}];

Formula

|T(n,k)| = A002260(n,k).

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0
Offset: 1

Views

Author

Robert G. Wilson v, Apr 27 2020

Keywords

Comments

\c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
r\
_0 0 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195 A067998
_1 0 1 1 0 -2 -5 -9 -14 -20 -27 -35 -44 -54 -65 -77 -90 A080956
_2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A001477
_3 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 A000217
_4 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 A000290
_5 0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 A000326
_6 0 1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 A000384
_7 0 1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 A000566
_8 0 1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 A000567
_9 0 1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 A001106
10 0 1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 A001107
11 0 1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 A051682
12 0 1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 A051624
13 0 1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 A051865
14 0 1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 A051866
15 0 1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

Links

Crossrefs

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Programs

  • Mathematica
    Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

Formula

P(r, c) = (r - 2)(c(c-1)/2) + c.
Previous Showing 31-35 of 35 results.

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