A023531 -id:A023531 - cp's OEIS frontend

A073423 Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.

2, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Jeremy Gardiner, Jul 30 2002

			T(2,1) = 0^2 + 0^0 = 1.
Triangle begins:
  2;
  1, 0;
  1, 0, 0;
  1, 0, 0, 0;
  1, 0, 0, 0, 0;
  1, 0, 0, 0, 0, 0;
  ...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Conjecture 4.9, p. 138.

Cf. A023531, A010054, A073424.

Column k=0 gives A054977.

A073423(n) = if(!n,2,ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

from math import isqrt
def A073423(n): return int((k:=n<<1)==(m:=isqrt(k))*(m+1)) if n else 2 # Chai Wah Wu, Nov 09 2024

a(0) = 2; and for n > 0, a(n) = A010054(n). [As a flat sequence] - Antti Karttunen, Jan 19 2025

A118011 Complement of the Connell sequence (A001614); a(n) = 4*n - A001614(n).

Original entry on oeis.org

3, 6, 8, 11, 13, 15, 18, 20, 22, 24, 27, 29, 31, 33, 35, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 76, 78, 80, 83, 85, 87, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 1

Paul D. Hanna, Apr 10 2006

nonn

a(n) is the position of the second appearance of n in A117384, where A117384(m) = A117384(k) and k = 4*A117384(m) - m. The Connell sequence (A001614) is generated as: 1 odd, 2 even, 3 odd, ...

Cf. A001614, A014132, A023531, A117384, A118012.

A171152 gives partial sums.

[2*n+Round(Sqrt(2*n)): n in [1..70]]; // Vincenzo Librandi, Apr 16 2015

Table[2 n + Round[Sqrt[2 n]], {n, 70}] (* Vincenzo Librandi, Apr 16 2015 *)

from math import isqrt
def A118011(n): return (m:=n<<1)+(k:=isqrt(m))+int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

A001614(n) = A118012(a(n)).
a(n) = 2n+[(1+sqrt(8n-7))/2]. - Juri-Stepan Gerasimov Aug 25 2009
a(n) = 2*n+round(sqrt(2*n)). - Gerald Hillier, Apr 16 2015
From Robert Israel, Apr 20 2015 (Start):
a(n) = 2*n + 1 + Sum_{j=0..n-2} A023531(j).
G.f.: 2*x/(1-x)^2 + x/(1-x) * Sum_{j=0..oo} x^(j*(j+1)/2) = 2*x/(1-x)^2 + x^(7/8)/(2-2*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)
a(n) = n+A014132(n). - Chai Wah Wu, Oct 19 2024

A007604 a(n) = a(n-1) + a(n-1-(number of odd terms so far)).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 16, 22, 31, 40, 52, 68, 90, 121, 152, 192, 244, 312, 402, 523, 644, 796, 988, 1232, 1544, 1946, 2469, 2992, 3636, 4432, 5420, 6652, 8196, 10142, 12611, 15080, 18072, 21708, 26140, 31560, 38212, 46408, 56550, 69161, 81772, 96852

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

A003056(n) gives the number of odd terms in the first n terms of this sequence. Modulo 2, this sequence becomes A023531. - T. D. Noe, Jul 24 2007

The present definition was the original definition of this sequence. It was later changed to "Sequence formed from rows of triangle A046936", but this seems less satisfactory. - N. J. A. Sloane, Oct 26 2014

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

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A006336, A046936, A003056.

a007604 n = a007604_list !! (n-1)
a007604_list = concat $ map tail $ tail a046936_tabl
-- Reinhard Zumkeller, Jan 01 2014

A[1]:= 1: A[2]:= 2: o:= 1:
for n from 3 to 100 do
  A[n]:= A[n-1] + A[n-1-o];
  if A[n]::odd then o:= o+1 fi
od:
seq(A[i],i=1..100); # Robert Israel, Mar 14 2023

a[n_Integer] := a[n] = Block[{c, k}, c = 0; k = 1; While[k < n, If[ OddQ[ a[k] ], c++ ]; k++ ]; Return[a[n - 1] + a[n - 1 - c] ] ]; a[1] = 1; a[2] = 2; Table[ a[n], {n, 0, 60} ]

Entry revised by N. J. A. Sloane, Oct 26 2014

A101508 Product of binomial matrix and the Mobius matrix A051731.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 3, 1, 16, 8, 6, 4, 1, 32, 16, 11, 10, 5, 1, 64, 32, 21, 20, 15, 6, 1, 128, 64, 42, 36, 35, 21, 7, 1, 256, 128, 85, 64, 70, 56, 28, 8, 1, 512, 256, 171, 120, 127, 126, 84, 36, 9, 1, 1024, 512, 342, 240, 220, 252, 210, 120, 45, 10, 1, 2048, 1024, 683, 496, 385, 463, 462, 330, 165, 55, 11, 1

Offset: 0

Paul Barry, Dec 05 2004

Row sums are A101509. Diagonal sums are A101510.
The matrix inverse appears to be A128313. - R. J. Mathar, Mar 22 2013
Read as upper triangular matrix, this can be seen as "recurrences in A135356 applied to A023531" [Paul Curtz, Mar 03 2017]. - The columns are: A000079, A131577, A024495, A000749, A139761, ... Column n differs after the (n+1)-th nonzero term on from the binomial coefficients C(k,n). - M. F. Hasler, Mar 05 2017

			Rows begin
  1;
  2,1;
  4,2,1;
  8,4,3,1;
  16,8,6,4,1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

A101508 := proc(n,k)
    a := 0 ;
    for i from 0 to n do
        if modp(i+1,k+1) = 0 then
            a := a+binomial(n,i) ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Mar 22 2013

t[n_, k_] := Sum[If[Mod[i + 1, k + 1] == 0, Binomial[n, i], 0], {i, 0, n}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *)

T(n,k)=sum(i=0,n, if((i+1)%(k+1)==0, binomial(n, i))) \\ M. F. Hasler, Mar 05 2017

T(n, k) = Sum_{i=0..n} if(mod(i+1, k+1)=0, binomial(n, i), 0).

Rows have g.f. x^k/((1-x)^(k+1)-x^(k+1)).

A173007 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n and q = 3.

Original entry on oeis.org

1, 3, 1, 27, 12, 1, 729, 351, 39, 1, 59049, 29160, 3510, 120, 1, 14348907, 7144929, 882090, 32670, 363, 1, 10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1, 22876792454961, 11433166050879, 1427185336941, 54665851779, 674887059, 2685501, 3279, 1

Offset: 0

Roger L. Bagula, Feb 07 2010

Triangle T(n,k), read by rows, given by [3,6,27,72,243,702,2187,6480,...] DELTA [1,0,3,0,9,0,27,0,81,0,243,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 01 2011

			Triangle begins as:
            1;
            3,          1;
           27,         12,         1;
          729,        351,        39,        1;
        59049,      29160,      3510,      120,      1;
     14348907,    7144929,    882090,    32670,    363,    1;
  10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), this sequence (q=3), A173008 (q=4).

Cf. A203148, A290000.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,3): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

(* First program *)
p[x_, n_, q_] = If[n==0, 1, Product[x + q^i, {i, 1, n}]];
Table[CoefficientList[p[x, n, 3], x], {n, 0, 10}] (* modified by G. C. Greubel, Feb 20 2021 *)
(* Second program *)
T[n_, k_, q_]:= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

p(x,n,q) = 1 if n=0, Product_{i=1..n} (x + q^i) otherwise, with q=3.
T(n,k) = 3^n*T(n-1,k) + T(n-1,k-1), T(0,0)=1. - Philippe Deléham, Oct 01 2011
Sum_{k=0..n} T(n, k, 4) = A290000(n+1). - G. C. Greubel, Feb 20 2021

Edited by G. C. Greubel, Feb 20 2021

A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 3, 1, 3, 6, 1, 0, 15, 9, 1, 0, 18, 36, 12, 1, 0, 9, 81, 66, 15, 1, 0, 0, 108, 216, 105, 18, 1, 0, 0, 81, 459, 450, 153, 21, 1, 0, 0, 27, 648, 1305, 810, 210, 24, 1, 0, 0, 0, 594, 2673, 2970, 1323, 276, 27, 1, 0, 0, 0, 324, 3915, 7938

Offset: 1

Wolfdieter Lang

			{1}; {3,1}; {3,6,1}; {0,15,9,1}; {0,18,36,12,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, m) := s1(-2, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528.

Cf. A049348, A049404.

a(n, m) = 3*(3*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(2, m)).

A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 64, 56, 14, 1, 1024, 960, 280, 30, 1, 32768, 31744, 9920, 1240, 62, 1, 2097152, 2064384, 666624, 89280, 5208, 126, 1, 268435456, 266338304, 87392256, 12094464, 755904, 21336, 254, 1, 68719476736, 68451041280, 22638755840, 3183575040, 205605888, 6217920, 86360, 510, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...] = A014236 (first zero omitted) DELTA A077957 where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006

			Triangle begins:
      1;
      2,     1;
      8,     6,    1;
     64,    56,   14,    1;
   1024,   960,  280,   30,  1;
  32768, 31744, 9920, 1240, 62, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A023531 (q=0), A007318 (q=1), this sequence (q=2), A173007 (q=3), A173008 (q=4).

Cf. A006125, A022166, A028362.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,2): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

T[n_, k_, q_]:= T[n,k,q]= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

Sum_{k=0..n} T(n, k) = A028362(n).
T(n,0) = 2^(n*(n+1)/2) = A006125(n+1). - Philippe Deléham, Nov 05 2006
T(n,k) = 2^binomial(n+1-k,2) * A022166(n,k) for 0 <= k <= n. - Werner Schulte, Mar 25 2019

A147716 Triangle of coefficients in expansion of (14 + x)^n.

Original entry on oeis.org

1, 14, 1, 196, 28, 1, 2744, 588, 42, 1, 38416, 10976, 1176, 56, 1, 537824, 192080, 27440, 1960, 70, 1, 7529536, 3226944, 576240, 54880, 2940, 84, 1, 105413504, 52706752, 11294304, 1344560, 96040, 4116, 98, 1, 1475789056, 843308032, 210827008, 30118144, 2689120, 153664, 5488, 112, 1

Offset: 0

Philippe Deléham, Nov 11 2008

Triangle T(n,k), read by rows, given by [14, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Triangle begins :
       1;
      14,      1;
     196,     28,     1;
    2744,    588,    42,    1;
   38416,  10976,  1176,   56,  1;
  537824, 192080, 27440, 1960, 70, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001024, A023531, A130595.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), A038243 (q=5), A038255 (q=6), A027466 (q=7), A038279 (q=8), A038291 (q=9), A038303 (q=10), A038315 (q=11), A038327 (q=12), A133371 (q=13), this sequence (q=14), A027467 (q=15).

[14^(n-k)*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 15 2021

With[{m=8}, CoefficientList[CoefficientList[Series[1/(1-14*x-x*y), {x, 0, m}, {y, 0, m}], x], y]]//Flatten (* Georg Fischer, Feb 17 2020 *)

flatten([[14^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 15 2021

T(n,k) = binomial(n,k) * 14^(n-k).
G.f.: 1/(1 - 14*x - x*y). - R. J. Mathar, Aug 12 2015
Sum_{k=0..n} T(n, k) = 15^n = A001024(n). - G. C. Greubel, May 15 2021

a(36) corrected by Georg Fischer, Feb 17 2020

A173008 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n, column 0<=k<=n, and q = 4.

Original entry on oeis.org

1, 4, 1, 64, 20, 1, 4096, 1344, 84, 1, 1048576, 348160, 22848, 340, 1, 1073741824, 357564416, 23744512, 371008, 1364, 1, 4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1, 72057594037927936, 24017731997138944, 1600791219535872, 25384570585088, 99158478848, 95414592, 21844, 1

Offset: 0

Roger L. Bagula, Feb 07 2010

Row sums are 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, ... (partial products of A092896).

Triangle T(n,k), read by rows, given by [4,12,64,240,1024,4032,16384,...] DELTA [1,0,4,0,16,0,64,0,256,0,1024,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 01 2011

			Triangle begins as:
              1;
              4,             1;
             64,            20,           1;
           4096,          1344,          84,          1;
        1048576,        348160,       22848,        340,       1;
     1073741824,     357564416,    23744512,     371008,    1364,    1;
  4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1;

Robert Israel, Table of n, a(n) for n = 0..1430

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), A173007 (q=3), this sequence (q=4).

Cf. A092896, A108084, A309327.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,4): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

P:= 1: A:= 1:
for n from 1 to 12 do
  P:= expand(P*(x+4^n));
  A:= A, seq(coeff(P,x,j),j=0..n)
od:
A; # Robert Israel, Aug 12 2015

(* First program *)
p[x_, n_, q_]= If[n==0, 1, Product[x + q^i, {i,n}]];
Table[CoefficientList[p[x, n, 4], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Feb 20 2021 *)
(* Second program *)
T[n_, k_, q_]:= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,4], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,4) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

T(n,k) = 4^n*T(n-1,k) + T(n-1,k-1) with T(0,0)=1. - Philippe Deléham, Oct 01 2011

Sum_{k=0..n} T(n, k, 4) = A309327(n+1). - G. C. Greubel, Feb 20 2021

A291203 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 3, 6, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 24, 12, 0, 36, 24, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 5, 80, 90, 20, 0, 200, 300, 60, 0, 300, 120, 0, 120, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 240, 540, 240, 30, 0, 1170, 3000, 1260, 120, 0, 3360, 2520, 360, 0, 2520, 720, 0, 720, 0

Offset: 0

Alois P. Heinz, Aug 20 2017

Positive elements in column t=1 give A034855.
Elements in rows h=0 give A023531.
Elements in rows h=1 give A059297.
Positive row sums per layer give A235595.
Positive column sums per layer give A061356.

			n h\t: 0   1   2  3  4 5 : A235595 : A061356          : A000272
-----+-------------------+---------+------------------+--------
0 0  : 1                 :         :                  : 1
-----+-------------------+---------+------------------+--------
1 0  : 0   1             :      1  :   .              :
1 1  : 0                 :         :   1              : 1
-----+-------------------+---------+------------------+--------
2 0  : 0   0   1         :      1  :   .   .          :
2 1  : 0   2             :      2  :   .              :
2 2  : 0                 :         :   2   1          : 3
-----+-------------------+---------+------------------+--------
3 0  : 0   0   0  1      :      1  :   .   .   .      :
3 1  : 0   3   6         :      9  :   .   .          :
3 2  : 0   6             :      6  :   .              :
3 3  : 0                 :         :   9   6   1      : 16
-----+-------------------+---------+------------------+--------
4 0  : 0   0   0  0  1   :      1  :   .   .   .  .   :
4 1  : 0   4  24 12      :     40  :   .   .   .      :
4 2  : 0  36  24         :     60  :   .   .          :
4 3  : 0  24             :     24  :   .              :
4 4  : 0                 :         :  64  48  12  1   : 125
-----+-------------------+---------+------------------+--------
5 0  : 0   0   0  0  0 1 :      1  :   .   .   .  . . :
5 1  : 0   5  80 90 20   :    195  :   .   .   .  .   :
5 2  : 0 200 300 60      :    560  :   .   .   .      :
5 3  : 0 300 120         :    420  :   .   .          :
5 4  : 0 120             :    120  :   .              :
5 5  : 0                 :         : 625 500 150 20 1 : 1296
-----+-------------------+---------+------------------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.

b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
       binomial(n-1, j-1)*j*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))
    end:
g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);

b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[
     Binomial[n-1, j-1]*j*x^t*b[j-1, 0, h-1]*b[n-j, t, h], {j, 1, n}]]];
g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n - h}], {h, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 17 2022, after Alois P. Heinz *)

Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n).
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
F(2n,n,n) = A126804(n) for n>0.
F(n,0,n) = 1 = A000012(n).
F(n,1,1) = n = A001477(n) for n>1.
F(n,n-1,1) = n! = A000142(n) for n>0.
F(n,1,n-1) = A002378(n-1) for n>0.
F(n,2,1) = A000551(n).
F(n,3,1) = A000552(n).
F(n,4,1) = A000553(n).
F(n,1,2) = A001788(n-1) for n>2.
F(n,0,0) = A000007(n).

cp's OEIS Frontend

A073423 Sums of two powers of zero: triangle read by rows: T(m,n) = 0^n + 0^m, n >= 0, m = 0..n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A118011 Complement of the Connell sequence (A001614); a(n) = 4*n - A001614(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A007604 a(n) = a(n-1) + a(n-1-(number of odd terms so far)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

A101508 Product of binomial matrix and the Mobius matrix A051731.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

PARI

Formula

A173007 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n and q = 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma