A011971 -id:A011971 - cp's OEIS frontend

A095151 a(n+3) = 3a(n+2) - 2a(n+1) + 1 with a(0)=0, a(1)=2.

0, 2, 7, 18, 41, 88, 183, 374, 757, 1524, 3059, 6130, 12273, 24560, 49135, 98286, 196589, 393196, 786411, 1572842, 3145705, 6291432, 12582887, 25165798, 50331621, 100663268, 201326563, 402653154, 805306337, 1610612704, 3221225439

Offset: 0

Gary W. Adamson, May 30 2004

A sequence generated from a Bell difference row matrix, companion to A095150.
A095150 uses the same recursion rule but the multiplier [1 1 1] instead of [1 0 0].
For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012
For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012

			a(6) = 183 = 3*88 -2*41 + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Péter Burcsi, Gabriele Fici, Zsuzsanna Lipták, Rajeev Raman, and Joe Sawada, Generating a Gray code for prefix normal words in amortized polylogarithmic time per word, arXiv:2003.03222 [cs.DS], 2020.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000110, A011971, A095149, A095150.

List([0..30], n-> 3*2^n -(n+3)); # G. C. Greubel, Jul 26 2019

[3*2^n -(n+3): n in [0..30]]; // G. C. Greubel, Jul 26 2019

Table[3*2^n -(n+3), {n,0,30}] (* G. C. Greubel, Jul 26 2019 *)

vector(30, n, n--; 3*2^n -(n+3)) \\ G. C. Greubel, Jul 26 2019

[3*2^n -(n+3) for n in (0..30)] # G. C. Greubel, Jul 26 2019

Let M = a 3 X 3 matrix having Bell triangle difference terms (A095149 is composed of differences of the Bell triangle A011971): (fill in the 3 X 3 matrix with zeros): [1 0 0 / 1 1 0 / 2 1 2] = M. Then M^n * [1 0 0] = [1 n a(n)]. E.g. a(4) = 41 since M^4 * [1 0 0] = [1 4 41].
a(n) = 3*2^n -(n+3) = 2*a(n-1) + n +1 = A000295(n+2) - A000079(n). For n>0, a(n) = A077802(n). - Henry Bottomley, Oct 25 2004
From Colin Barker, Apr 23 2012: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: x*(2-x)/((1-x)^2*(1-2*x)). (End)
a(n) = A125128(n) + A000225(n). - Miquel Cerda, Aug 07 2016
a(n) = 2*A125128(n) - A000325(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = A125128(n) + A000325(n) + n - 1. - Miquel Cerda, Aug 27 2016
E.g.f.: 3*exp(2*x) - (3+x)*exp(x). - G. C. Greubel, Jul 26 2019
Let Prod_{i=0..n-1} (1+x^{2^i}+x^{2*2^i}) = Sum_{j=0..d} b_j x^j, where d=2^{n+1}-2. Then a(n) = Sum_{j=0..d-1} b_j/b_{j+1} (proved). - Richard Stanley, Aug 27 2019

Edited by Robert G. Wilson v, Jun 05 2004

Deleted a comment and file that were unrelated to this sequence. - N. J. A. Sloane, Aug 17 2025

A106436 Difference array of Bell numbers A000110 read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 3, 5, 4, 5, 7, 10, 15, 11, 15, 20, 27, 37, 52, 41, 52, 67, 87, 114, 151, 203, 162, 203, 255, 322, 409, 523, 674, 877, 715, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 3425, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Philippe Deléham, May 29 2005

Essentially Aitken's array A011971 with first column A000296.

Mirror image of A182930. - Alois P. Heinz, Jan 29 2019

			   1;
   0,  1;
   1,  1,  2;
   1,  2,  3,  5;
   4,  5,  7, 10, 15;
  11, 15, 20, 27, 37, 52;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A182930.
Diagonals give A005493, A011965-A011967, A191099, A000298, A011968-A011970.
T(2n,n) gives A020556.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
T:= proc(n, k) option remember; `if`(k=0, b(n),
      T(n+1, k-1)-T(n, k-1))
    end:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jan 29 2019

bb = Array[BellB, m = 12, 0];
dd[n_] := Differences[bb, n];
A = Array[dd, m, 0];
Table[A[[n-k+1, k+1]], {n, 0, m-1}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019 *)
a[0,0]:=1; a[n_,0]:=a[n-1,n-1]-a[n-1,0]; a[n_,k_]/;0Oliver Seipel, Nov 23 2024 *)

Double-exponential generating function: sum_{n, k} a(n-k, k) x^n/n! y^k/k! = exp(exp{x+y}-1-x). a(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,i-k)*Bell(i). - Vladeta Jovovic, Oct 14 2006

A129323 Second column of PE^2.

Original entry on oeis.org

0, 1, 4, 18, 88, 470, 2724, 17010, 113712, 809262, 6101820, 48540778, 405935688, 3557404838, 32577733972, 310987560930, 3087723669600, 31823217868318, 339845199259500, 3754422961010522, 42843681016834680, 504339820818380694

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A129323 := proc(n) A078937(n+1,1) ; end: seq(A129323(n),n=0..23) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 2], {i, 0, n}], {n, -1, 20}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,2] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,2]

More terms from R. J. Mathar, May 30 2008

A129324 Third column of PE^2.

Original entry on oeis.org

0, 0, 1, 6, 36, 220, 1410, 9534, 68040, 511704, 4046310, 33560010, 291244668, 2638581972, 24901833866, 244333004790, 2487900487440, 26245651191600, 286408960814862, 3228529392965250, 37544229610105220, 449858650676764140

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A129324 := proc(n) A078937(n+1,2) ; end: seq(A129324(n),n=0..23) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078937[n + 1, 2];
a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,3]; with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,3].

E.g.f.: (x^2/2) * exp(2 * (exp(x) - 1)). - Ilya Gutkovskiy, Jul 11 2020

More terms from R. J. Mathar, May 30 2008

A129325 Fourth column of PE^2.

Original entry on oeis.org

0, 0, 0, 1, 8, 60, 440, 3290, 25424, 204120, 1705680, 14836470, 134240040, 1262060228, 12313382536, 124509169330, 1303109358880, 14098102762160, 157473907149600, 1813923418494126, 21523529286435000, 262809607270736540

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A129325 := proc(n) A078937(n+1,3) ; end: seq(A129325(n),n=0..27) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078937[n + 1, 3];
a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

m=matpascal(30)-matid(31); pe=matid(31)+sum(i=1,30,m^i/i!); A=pe^2; A[,4] \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,4] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,4]

More terms from R. J. Mathar and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

A129327 Second column of PE^3.

Original entry on oeis.org

0, 1, 6, 36, 228, 1545, 11196, 86457, 708504, 6136830, 55976430, 535904259, 5369146272, 56145107577, 611336534802, 6916529431620, 81152874393168, 985767316792449, 12376996566040980, 160399065135692073

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129327 := proc(n) A078938(n+1,1) ; end: seq(A129327(n),n=0..27) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 3], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,2 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,2]

More terms from R. J. Mathar, May 30 2008

A129328 Third column of PE^3.

Original entry on oeis.org

0, 0, 1, 9, 72, 570, 4635, 39186, 345828, 3188268, 30684150, 307870365, 3215425554, 34899450768, 393015753039, 4585024011015, 55332235452960, 689799432341928, 8871905851132041, 117581467377389310, 1603990651356920730

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129328 := proc(n) A078938(n+1,2) ; end: seq(A129328(n),n=0..27) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078938[n + 1, 2];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,3 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,3]

More terms from R. J. Mathar, May 30 2008

A129329 Fourth column of PE^3.

Original entry on oeis.org

0, 0, 0, 1, 12, 120, 1140, 10815, 104496, 1037484, 10627560, 112508550, 1231481460, 13933510734, 162864103584, 1965078765195, 24453461392080, 313549334233440, 4138796594051568, 56188737057169593, 783876449182595400

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129329 := proc(n) A078938(n+1,3) ; end: seq(A129329(n),n=0..27) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078938[n + 1, 3];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,4 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,4]

E.g.f.: (x^3/6) * exp(3 * (exp(x) - 1)). - Ilya Gutkovskiy, Jul 11 2020

More terms from R. J. Mathar, May 30 2008

A129331 Second column of PE^4.

Original entry on oeis.org

0, 1, 8, 60, 464, 3780, 32568, 296492, 2845088, 28695060, 303334920, 3351877628, 38622668400, 463036981732, 5764038605528, 74365952622540, 992720923710272, 13690497077256628, 194777994524434344, 2855149354656290716

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129331 := proc(n) A078939(n+1,1) ; end: seq(A129331(n),n=0..25) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 4], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,2 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,2]

More terms from R. J. Mathar, May 30 2008

A129332 Third column of PE^4.

Original entry on oeis.org

0, 0, 1, 12, 120, 1160, 11340, 113988, 1185968, 12802896, 143475300, 1668342060, 20111265768, 251047344600, 3241258872124, 43230289541460, 594927620980320, 8438127851537312, 123214473695309652, 1850390947982126268

Offset: 0

Gottfried Helms, Apr 08 2007

Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129332 := proc(n) A078939(n+1,2) ; end: seq(A129332(n),n=0..25) ; # R. J. Mathar, May 30 2008

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
A078939[n_, c_] := Sum[A078938[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078939[n + 1, 2];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,3 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,3]

More terms from R. J. Mathar, May 30 2008

cp's OEIS Frontend

A095151 a(n+3) = 3*a(n+2) - 2*a(n+1) + 1 with a(0)=0, a(1)=2.

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A106436 Difference array of Bell numbers A000110 read by antidiagonals.

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A129323 Second column of PE^2.

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A129324 Third column of PE^2.

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A129325 Fourth column of PE^2.

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A129327 Second column of PE^3.

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A129328 Third column of PE^3.

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A095151 a(n+3) = 3a(n+2) - 2a(n+1) + 1 with a(0)=0, a(1)=2.