A125860 -id:A125860 - cp's OEIS frontend

A125859 Antidiagonal sums of table A125860.

1, 2, 4, 10, 35, 184, 1531, 21080, 497017, 20533486, 1508839043, 199272672334, 47686000150774, 20817464210086523, 16678749474397158418, 24657143458135746104239, 67591557017940565183386368

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Cf. A125860; A016121, A125862, A125863, A125864, A125865, A125861.

A125861 Main diagonal of table A125860.

Original entry on oeis.org

1, 2, 12, 178, 5990, 435801, 66593821, 20997402098, 13512727916532, 17629371074833300, 46432767742317108086, 246240366959004185679198, 2624854986865673643625591411, 56179604057909797695704800461149

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Cf. A125860; A097712; columns: A016121, A125862, A125863, A125864, A125865; A125859 (antidiagonal sums).

A125862 Column 2 of table A125860; also equals row sums of matrix power A097712^2.

Original entry on oeis.org

1, 3, 12, 69, 612, 8853, 217041, 9245253, 695682342, 93580638024, 22713677612832, 10025101876435413, 8100572528598910191, 12054728928174188426943, 33214476295395054879355617, 170255688895444623691322464599

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Equals column 0 of matrix power A097712^3, where triangle A097712 satisfies recurrence: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1).

Cf. A125860; A097712; other columns: A016121, A125863, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums).

A125863 Column 3 of table A125860; also equals row sums of matrix power A097712^3.

Original entry on oeis.org

1, 4, 22, 178, 2251, 46663, 1640572, 100152049, 10794383587, 2079805452133, 723492192295786, 458149292979837523, 531871667833026397222, 1138955362720160687114704, 4523369812874327770490887837

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Equals column 0 of matrix power A097712^4, where triangle A097712 satisfies recurrence: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1).

Cf. A125860; A097712; other columns: A016121, A125862, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums).

A125864 Column 4 of table A125860; also equals row sums of matrix power A097712^4.

Original entry on oeis.org

1, 5, 35, 365, 5990, 161525, 7387640, 586285040, 82090572095, 20540291522675, 9278896006526795, 7632398133742637255, 11514756687812563119530, 32063466203746720003813970, 165699104606274900865952221145

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Equals column 0 of matrix power A097712^5, where triangle A097712 satisfies recurrence: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1).

Cf. A125860; A097712; other columns: A016121, A125862, A125863, A125865; A125861 (diagonal), A125859 (antidiagonal sums).

A125865 Column 5 of table A125860; also equals row sums of matrix power A097712^5.

Original entry on oeis.org

1, 6, 51, 651, 13131, 435801, 24530016, 2394413286, 412135908606, 126722253316281, 70336222713070656, 71088278975389204986, 131802456226253519662956, 451158681817567800972000111

Offset: 0

Paul D. Hanna, Dec 13 2006

nonn

Equals column 0 of matrix power A097712^6, where triangle A097712 satisfies recurrence: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1).

Cf. A125860; A097712; other columns: A016121, A125862, A125863, A125864; A125861 (diagonal), A125859 (antidiagonal sums).

A005945 Number of n-step mappings with 4 inputs.

Original entry on oeis.org

0, 1, 15, 60, 154, 315, 561, 910, 1380, 1989, 2755, 3696, 4830, 6175, 7749, 9570, 11656, 14025, 16695, 19684, 23010, 26691, 30745, 35190, 40044, 45325, 51051, 57240, 63910, 71079, 78765, 86986, 95760, 105105, 115039, 125580, 136746

Offset: 0

N. J. A. Sloane

a(n) is the coefficient of x^4/4! in n-th iteration of exp(x)-1.

			G.f. = x + 15*x^2 + 60*x^3 + 154*x^4 + 315*x^5 + 561*x^6 + 910*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
B. A. Huberman, T. H. Hogg, & N. J. A. Sloane, Correspondence, 1985
Pierpaolo Natalini, Paolo E. Ricci, Integer Sequences Connected with Extensions of the Bell Polynomials, Journal of Integer Sequences, 2017, Vol. 20, #17.10.2.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005920, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. [Bruno Berselli, Apr 25 2010]

Cf. A094952.

I:=[0, 1, 15, 60]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi Jun 18 2012

LinearRecurrence[{4,-6,4,-1},{0,1,15,60},50] (* Vincenzo Librandi, Jun 18 2012 *)
a[ n_] := 3 n^3 - 5/2 n^2 + 1/2 n; (* Michael Somos, Jun 10 2015 *)

{a(n) = 3*n^3 - 5/2*n^2 + 1/2*n}; /* Michael Somos, Jan 23 2014 */

G.f.: x*(1+11*x+6*x^2)/(1-x)^4. a(n)=n*(3*n-1)*(2*n-1)/2.
For n>0, a(n) = n*A000567(n) - A000217(n-1). - Bruno Berselli, Apr 25 2010; Feb 01 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 18 2012
a(n) = -A094952(-n) for all n in Z. - Michael Somos, Jan 23 2014

Edited by Michael Somos, Oct 29 2002

A005920 Tricapped prism numbers.

Original entry on oeis.org

1, 9, 33, 82, 165, 291, 469, 708, 1017, 1405, 1881, 2454, 3133, 3927, 4845, 5896, 7089, 8433, 9937, 11610, 13461, 15499, 17733, 20172, 22825, 25701, 28809, 32158, 35757, 39615, 43741, 48144, 52833, 57817, 63105, 68706, 74629, 80883, 87477, 94420

Offset: 0

N. J. A. Sloane

a(n) = (n+1)*A000326(n+1) - Sum_{i=0...n} A001477(i) = (n+1)*((n+1)*(3*n+2)/2) - A000217(n) = (n+1)*(3*n^2+4n+2)/2. - Bruno Berselli, Apr 25 2010

Also central terms of triangle A093445: a(n) = A093445(2*n+1,n+1). - Reinhard Zumkeller, Oct 03 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005945, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. - Bruno Berselli, Apr 25 2010

Cf. A000326, A001477, A093445.

a005920 n = (n * (n * (3 * n + 7) + 6) + 2) `div` 2
-- Reinhard Zumkeller, Oct 03 2012

[(3*n^3+7*n^2+6*n+2)/2 : n in [0..50]]; // Wesley Ivan Hurt, May 05 2021

a:=n->(3*n^3+7*n^2+6*n+2)/2: seq(a(n),n=0..60);
A005920:=(1+5*z+3*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[ Series[(1+5x+3x^2)/(1-x)^4, {x, 0, 39}], x] (* Jean-François Alcover, Dec 02 2011, after Simon Plouffe *)
LinearRecurrence[{4,-6,4,-1},{1,9,33,82},40] (* Harvey P. Dale, Sep 25 2012 *)

a(n)=n*(3*n^2+7*n+6)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (1/2) * (3*n^3 + 7*n^2 + 6*n + 2). - Ralf Stephan, Apr 20 2004
a(0)=1, a(1)=9, a(2)=33, a(3)=82, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Sep 25 2012
E.g.f.: exp(x)*(2 + 16*x + 16*x^2 + 3*x^3)/2. - Stefano Spezia, Jun 10 2022

More terms from Emeric Deutsch, May 09 2004

cp's OEIS Frontend

A125859 Antidiagonal sums of table A125860.

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A125861 Main diagonal of table A125860.

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A125862 Column 2 of table A125860; also equals row sums of matrix power A097712^2.

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A125863 Column 3 of table A125860; also equals row sums of matrix power A097712^3.

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A125864 Column 4 of table A125860; also equals row sums of matrix power A097712^4.

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A125865 Column 5 of table A125860; also equals row sums of matrix power A097712^5.

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A005945 Number of n-step mappings with 4 inputs.

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A005920 Tricapped prism numbers.

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