A101096 -id:A101096 - cp's OEIS frontend

A101100 The first summation of row 5 of Euler's triangle - a row that will recursively accumulate to the power of 5.

1, 27, 93, 119, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 533.

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.
Eric Weisstein's World of Mathematics Worpitzky's Identity of 1883.
Eric Weisstein's World of Mathematics Eulerian Number.
Eric Weisstein's World of Mathematics Nexus number.
Eric Weisstein's World of Mathematics Finite Differences.
Index entries for linear recurrences with constant coefficients, signature (1).

Within the "cube" of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward: Before: this sequence, A101095, A101096, A101098, A022521, A000584, A000539, A101092, A101099. Above: A101104, this sequence.

Within the "cube" of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward and r backward: Before: this sequence, A101095, A101096, A101098, A022521, A000584, A000539, A101092, A101099.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x) )); // G. C. Greubel, May 07 2019

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 5, 5}, {z, 1, 1}, {k, 0, 34}]
(* or *)
SeriesAtLevelR = Sum[Eulerian[n, i-1]*Binomial[n+x-i+r, n+r], {i,1,n}]; Table[SeriesAtLevelR, {n, 5, 5}, {r, -5, -5}, {x, 5, 35}]

{a(n) = if(n==1, 1, if(n==2, 27, if(n==3, 93, if(n==4, 119, 120))) )}; \\ G. C. Greubel, May 07 2019

a=(x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 07 2019

a(n) = 120, n>4.
a(n) = Sum_{j=1..m} Eulerian(m, j-1)*binomial(m+n-j+r, m+r), with m = 5, r = -5.
a(n) = Sum_{j=0..n+1} (-1)^j*binomial(m+1-z, j)*(n-j+1)^n, with m = 5, z = 1.
G.f.: x*(1+26*x+66*x^2+26*x^3+x^4)/(1-x). - Colin Barker, Mar 01 2012

A101101 a(1)=1, a(2)=5, and a(n)=6 for n >= 3.

Original entry on oeis.org

1, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Previous name was: The first summation of row 3 of Euler's triangle - a row that will recursively accumulate to the power of 3.

Decimal expansion of 47/30. - Elmo R. Oliveira, Aug 09 2024

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al.), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [broken link: domain now owned by a domain grabber]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883.
Eric Weisstein, Link to section of MathWorld: Eulerian Number.
Eric Weisstein, Link to section of MathWorld: Nexus number.
Eric Weisstein, Link to section of MathWorld: Finite Differences.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Within the "cube" of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward:
Before: this_sequence, A008458, A003215, A000578, A000537, A024166 or A024166, A101094, A101097, A101102.
Above: this_sequence, below: A101104, A101100.
Within the "cube" of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward and r backward:
Above: this_sequence, below: A101103, A101096.

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 3, 3}, {z, 1, 1}, {k, 0, 34}] (* OR *)
SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 3, 3}, {r, -3, -3}, {x, 4, 35}]
Join[{1, 5},LinearRecurrence[{1},{6},78]] (* Ray Chandler, Sep 23 2015 *)

G.f.: x*(1+4*x+x^2)/(1-x). - L. Edson Jeffery, Jan 29 2012

I wish the sequence was as interesting as the list of references! - N. J. A. Sloane

New name from Joerg Arndt, Nov 30 2014

A069477 a(n) = 60n^2 + 180n + 150.

Original entry on oeis.org

390, 750, 1230, 1830, 2550, 3390, 4350, 5430, 6630, 7950, 9390, 10950, 12630, 14430, 16350, 18390, 20550, 22830, 25230, 27750, 30390, 33150, 36030, 39030, 42150, 45390, 48750, 52230, 55830, 59550, 63390, 67350, 71430, 75630, 79950, 84390, 88950, 93630, 98430, 103350

Offset: 1

Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Apr 11 2002

First differences of A068236, successive differences of (n+1)^5 - n^5 (A022521).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001844, A022521, A068236, A069477, A069894, A101096.

[30*(2*n^2 + 6*n + 5): n in [1..40]]; // Vincenzo Librandi, Nov 23 2011

Table[30 (2 n^2 + 6 n + 5), {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2011 *)
LinearRecurrence[{3,-3,1},{390,750,1230},40] (* Harvey P. Dale, Apr 06 2012 *)

a(n)=60*n^2+180*n+150 \\ Charles R Greathouse IV, Nov 23 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=390, a(2)=750, a(3)=1230. - Harvey P. Dale, Apr 06 2012
Sum_{n>=1} 1/a(n) = (Pi/60)*tanh(Pi/2) - 1/25. - Amiram Eldar, Jan 27 2022
From Elmo R. Oliveira, Feb 08 2025: (Start)
G.f.: 30*x*(5*x^2 - 14*x + 13)/(1-x)^3.
E.g.f.: 30*(exp(x)*(2*x^2 + 8*x + 5) - 5).
a(n) = 30*A001844(n+1) = 15*A069894(n+1). (End)

cp's OEIS Frontend

A101100 The first summation of row 5 of Euler's triangle - a row that will recursively accumulate to the power of 5.

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A101101 a(1)=1, a(2)=5, and a(n)=6 for n >= 3.

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A069477 a(n) = 60n^2 + 180n + 150.

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cp's OEIS Frontend

A101100 The first summation of row 5 of Euler's triangle - a row that will recursively accumulate to the power of 5.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A101101 a(1)=1, a(2)=5, and a(n)=6 for n >= 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A069477 a(n) = 60*n^2 + 180*n + 150.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A069477 a(n) = 60n^2 + 180n + 150.