A136431 -id:A136431 - cp's OEIS frontend

A176085 a(n) = A136431(n,n).

0, 1, 3, 11, 41, 155, 591, 2267, 8735, 33775, 130965, 509015, 1982269, 7732659, 30208749, 118167055, 462760369, 1814091011, 7118044023, 27952660883, 109853552255, 432021606103, 1700093447847, 6694137523051, 26372544576331, 103950885100775, 409928481296331

Offset: 0

Paul Curtz, Apr 08 2010

a(n+1) is also the number of sequences of length 2n obeying the regular expression "0^* (1 or 2)^* 3^*" and having sum 3n. For example, a(3)=11 because of the sequences 0033, 0123, 0213, 0222, 1113, 1122, 1212, 1221, 2112, 2121, 2211. - Don Knuth, May 11 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000045, A054441, A108617.

List([0..30], n-> Sum([0..n], k-> Binomial(2*n-k-1, n-k)*Fibonacci(k) )); # G. C. Greubel, Nov 28 2019

[(&+[Binomial(2*n-k-1, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 28 2019

with(combinat); seq( add(binomial(2*n-k-1, n-k)*fibonacci(k), k=0..n), n=0..30); # G. C. Greubel, Nov 28 2019
1/(sqrt(1 - 4*x) + 1/x - 4): series(%, x, 27):
seq(coeff(%, x, k), k=0..26); # Peter Luschny, May 29 2021

t[n_, k_]:= CoefficientList[ Series[x/(1-x-x^2)/(1-x)^k, {x,0,k}], x][[k+1]]; Array[ t[#, #] &, 20]
Table[Sum[Binomial[2*n-k-1, n-k]*Fibonacci[k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Nov 28 2019 *)

a(n):=sum(fib(k)*binomial(2*n-k-1,n-k),k,1,n); /*  Vladimir Kruchinin, Mar 17 2016 */

a(n) = sum(k=1, n, fibonacci(k)*binomial(2*n-k-1, n-k)) \\ Michel Marcus, Mar 17 2016

[sum(binomial(2*n-k-1, n-k)*fibonacci(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 28 2019

a(n+1) - 4*a(n) = -A081696(n-1).
From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: x*(x-sqrt(1-4*x))/(sqrt(1-4*x)*(x^2+4*x-1)).
Recurrence: (n-2)*a(n) = 2*(4*n-9)*a(n-1) - (15*n-38)*a(n-2) - 2*(2*n-5)* a(n-3).
a(n) ~ 4^n/sqrt(Pi*n). (End)
a(n) = Sum_{k=1..n} (F(k)*binomial(2*n-k-1,n-k)), where F(k) = A000045(k). - Vladimir Kruchinin, Mar 17 2016
Simpler g.f.: x/sqrt(1-4*x)/(x+sqrt(1-4*x)). - Don Knuth, May 11 2016
a(n) = A000045(3*n) - A054441(n). - Hrishikesh Venkataraman, May 27 2021
a(n) = 4*a(n-1) + a(n-2) - binomial(2*n-4,n-2) for n>=2. - Hrishikesh Venkataraman, Jul 02 2021
a(n) = A108617(2n,n)/2. - Alois P. Heinz, Jan 26 2025

A136338 Primes in the array A136431 that are not Fibonacci numbers.

Original entry on oeis.org

7, 11, 29, 37, 41, 67, 79, 97, 137, 191, 211, 277, 379, 631, 709, 821, 947, 967, 991, 1129, 1327, 1597, 1831, 2017, 2081, 2267, 2347, 2557, 2683, 2851, 2927, 3571, 3917, 4561, 4657, 4951, 5051, 5779, 6217, 6329, 6763, 8273, 8647, 8779, 9181, 9871, 10093

Offset: 1

Jonathan Vos Post, Apr 12 2008

A generalization of prime Fibonacci numbers (A005478) are the prime hyperfibonacci numbers (primes in A136431). Referring to the array A(k,n) = Apply partial sum operator k times to Fibonacci numbers, we see that every prime occurs in the n=2 column (as it contains every positive integer).
So excluding n=2 and k=0 (A005478) we have the nontrivially prime hyperfibonacci numbers which are not Fibonacci numbers.
Note that this sequence does not indicate multiplicity (e.g., 7 occurs twice in the valid part of the table).
Continuing the table of primes in the examples, from a computation by Joshua Zucker, we have:
k=1: {7, ...} no more through n = 1000.
k=2: {7, 79, 514201, 14930317, 956722025983, 5527939700884681 4660046610375530219, ...}
k=3: {11, 97, 17519, next value has 60 digits, ...}
k=4: {41, 10093, 16703, 3520457, 591286703533, 6557470285501, 19740274219868101499, ...}
k=5: {709, 8273, 14323, 466004661037329684,1 298611126818977061133263, ...}
k=6: {29, 2683, 23945893, 1835540197, 4052735290427, 27777884012083, ...}
k=7: {37, 967, 2267, 127921, 226007, 62048869, 1131463777, 7540113804271826929, ...}
k=8: {27777538280521, 1409869790947669143312035590804646728957, ...}
k=9: {1033628323428189498226451492123369099, next value has 60 digits, ...}
k=10: {67, 5972304273877744135569337875802249660927, ...}
k=11: {79, 4478413, 19008291293, 61305228407581679, ...}
k=12: {6763, 1982269, 37886753582095837, 2791715456569622316696636389, ...}.

			k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 1000.
k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79, A001924(25) = 514201, ...}
k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97, A014162(16) = 17519}, no more through n = 30.
k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093, A014166(14) = 16703}
k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273, A053739(11) = 14323}, no more through n = 27.
k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683, 23945893(24) = 23945893}, no more through n = 27.
k=7: primes in A053296 = {A053296(3) = 37, A053296(6) = 967, A053296(7) = 2267, A053296(12) = 127921, A053296(13) = 226007}, no more through n = 27.

Cf. A000040, A005478, A136431, A137176.

Cf. A136431.

A136431 := proc(k,n) local x ; coeftayl(x/(1-x-x^2)/(1-x)^k,x=0,n) ; end: A136338 := proc(amax) local a,k,n,a136431; a := [] ; for k from 1 do if A136431(k,3) > amax then break ; fi ; for n from 3 do a136431 := A136431(k,n) ; if a136431 > amax then break ; fi ; if isprime(a136431) and not a136431 in a then a := [op(a),a136431] ; fi ; od: od: sort(a) ; end: A136338(20000) ; # R. J. Mathar, Apr 21 2008

partsumfib(N,s=[],P=[])={ for( n=1+#s,N, s=concat(s,n+1); forstep( i=n,1,-1, isprime( s[i]+= if( i>1, s[i-1], fibonacci(n+2) ) ) & P=setunion(P,[s[i]]) ); print(s); );vecsort(eval(P))} \\ M. F. Hasler

Primes in the hyperfibonacci number array of A136431, excluding the n=2 column (which contains every positive integer).

Revised definition from N. J. A. Sloane, May 09 2008

More terms from R. J. Mathar, Apr 21 2008

A162741 Fibonacci-Pascal triangle; same as Pascal triangle, but beginning another Pascal triangle to the right of each row starting at row 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 7, 7, 5, 3, 2, 1, 1, 1, 5, 11, 14, 12, 8, 5, 3, 2, 1, 1, 1, 6, 16, 25, 26, 20, 13, 8, 5, 3, 2, 1, 1, 1, 7, 22, 41, 51, 46, 33, 21, 13, 8, 5, 3, 2, 1, 1, 1, 8, 29, 63, 92, 97, 79, 54, 34, 21, 13, 8, 5, 3, 2, 1, 1

Offset: 1

Mark Dols, Jul 12 2009, Jul 19 2009

Intertwined Pascal-triangles;

the first five rows seen as numbers in decimal representation: row(n) = 110*row(n-1) + 1. - corrected by Reinhard Zumkeller, Jul 16 2013

			.                                           1
.                                       1,  1, 1
.                                   1,  2,  2, 1, 1
.                               1,  3,  4,  3, 2, 1, 1
.                           1,  4,  7,  7,  5, 3, 2, 1, 1
.                       1,  5, 11, 14, 12,  8, 5, 3, 2, 1, 1
.                   1,  6, 16, 25, 26, 20, 13, 8, 5, 3, 2, 1,1
.               1,  7, 22, 41, 51, 46, 33, 21,13, 8, 5, 3, 2,1,1
.           1,  8, 29, 63, 92, 97, 79, 54, 34,21,13, 8, 5, 3,2,1,1
.       1,  9, 37, 92,155,189,176,133, 88, 55,34,21,13, 8, 5,3,2,1,1
.    1,10, 46,129,247,344,365,309,221,143, 89,55,34,21,13, 8,5,3,2,1,1
. 1,11,56,175,376,591,709,674,530,364,232,144,89,55,34,21,13,8,5,3,2,1,1 .

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Table 3.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A005408 (row length), A000225 (row sums), A000045 (central terms), A007318, A136431.
Cf. A021113. - Mark Dols, Jul 18 2009
Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A228074.

a162741 n k = a162741_tabf !! (n-1) !! (k-1)
a162741_row n = a162741_tabf !! (n-1)
a162741_tabf = iterate
   (\row -> zipWith (+) ([0] ++ row ++ [0]) (row ++ [0,1])) [1]
-- Reinhard Zumkeller, Jul 16 2013

T[, 1] = 1; T[n, k_] /; k == 2*n-2 || k == 2*n-1 = 1; T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k]; Table[T[n, k], {n, 1, 9}, {k, 1, 2*n-1}] // Flatten (* Jean-François Alcover, Oct 30 2017, after Reinhard Zumkeller *)

T(n,k) = T(n-1,k-1) + T(n-1,k), T(n,1)=1 and for n>1: T(n,2*n-2) = T(n,2*n-1)=1. - Reinhard Zumkeller, Jul 16 2013

A014166 Apply partial sum operator 4 times to Fibonacci numbers.

Original entry on oeis.org

0, 1, 5, 16, 41, 92, 189, 365, 674, 1204, 2098, 3588, 6050, 10093, 16703, 27476, 44995, 73440, 119575, 194345, 315460, 511576, 829060, 1342936, 2174596, 3520457, 5698329, 9222440, 14924829, 24151764, 39081553

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hung Viet Chu, Partial Sums of the Fibonacci Sequence, arXiv:2106.03659 [math.CO], 2021.
Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, arXiv:1606.06228 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A000045, A228074, A136431.

Right-hand column 8 of triangle A011794.

List([0..30], n-> Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6); # G. C. Greubel, Sep 06 2019

[Fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6: n in [0..30]]; // G. C. Greubel, Sep 06 2019

with(combinat); seq(fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6, n = 0..30); # G. C. Greubel, Sep 06 2019

Nest[Accumulate, Fibonacci[Range[0, 30]], 4] (* Jean-François Alcover, Jan 08 2019 *)

a(n)=fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6 \\ Charles R Greathouse IV, Jun 11 2015

[fibonacci(n+8)-(n^3+12*n^2+59*n+126)/6 for n in (0..30)] # G. C. Greubel, Sep 06 2019

a(n) = Fibonacci(n+8) - (n^3 +12*n^2 +59*n +126)/6.

G.f.: x/((1-x)^4*(1-x-x^2)).

A053296 Partial sums of A053295.

Original entry on oeis.org

1, 8, 37, 129, 376, 967, 2267, 4950, 10220, 20175, 38403, 70954, 127921, 226007, 392688, 672959, 1140260, 1914166, 3189022, 5280288, 8699540, 14275838, 23352118, 38102976, 62048869, 100888126, 163843187, 265838881, 431026972, 698489013, 1131463777, 1832277574, 2966502032, 4802042229

Offset: 0

Barry E. Williams, Mar 04 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-27,49,-49,21,7,-13,6,-1).

Cf. A000045, A053739, A014166, A136431, A228074.

Right-hand column 14 of triangle A011794.

[(&+[Binomial(n+7-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018

Table[Sum[Binomial[n+7-j, n-2*j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)

for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+7-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

a(n) = Sum_{i=0..floor(n/2)} C(n+7-i, n-2i), n >= 0.
a(n) = a(n-1) + a(n-2) + C(n+6,6); n >= 0, with a(-1) = 0.
From G. C. Greubel, Oct 21 2024: (Start)
a(n) = Fibonacci(n+15) - Sum_{j=0..6} Fibonacci(14-2*j)*binomial(n+j,j).
a(n) = Fibonacci(n+15) - (1/6!)*(n^6 + 39*n^5 + 685*n^4 + 7185*n^3 + 48994*n^2 + 209496*n + 438480).
G.f.: 1/((1-x)^7*(1 - x - x^2)). (End)

Terms a(28) onward added by G. C. Greubel, May 24 2018

A053308 Partial sums of A053296.

Original entry on oeis.org

1, 9, 46, 175, 551, 1518, 3785, 8735, 18955, 39130, 77533, 148487, 276408, 502415, 895103, 1568062, 2708322, 4622488, 7811510, 13091798, 21791338, 36067176, 59419294, 97522270, 159571139, 260459265, 424302452, 690141333

Offset: 0

Barry E. Williams, Mar 06 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-35,76,-98,70,-14,-20,19,-7,1).

Cf. A053296, A053295, A136431.

Cf. A228074.

[(&+[Binomial(n+8-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018

Table[Sum[Binomial[n+8-j, n-2j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)

for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+8-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

a(n) = Sum_{i=0..floor(n/2)} C(n+8-i, n-2i), n >= 0.

a(n) = a(n-1) + a(n-2) + C(n+7,7); n >= 0; a(-1)=0.

A053309 Partial sums of A053308.

Original entry on oeis.org

1, 10, 56, 231, 782, 2300, 6085, 14820, 33775, 72905, 150438, 298925, 575333, 1077748, 1972851, 3540913, 6249235, 10871723, 18683233, 31775031, 53566369, 89633545, 149052839, 246575109, 406146248, 666605513, 1090907965

Offset: 0

Barry E. Williams, Mar 06 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-44,111,-174,168,-84,-6,39,-26,8,-1).

Cf. A053296, A053295, A136431.

Cf. A228074.

[(&+[Binomial(n+9-j, n-2*j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 24 2018

Table[Sum[Binomial[n+9-j, n-2j], {j, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, May 24 2018 *)

for(n=0, 30, print1(sum(j=0, floor(n/2), binomial(n+9-j, n-2*j)), ", ")) \\ G. C. Greubel, May 24 2018

a(n) = Sum_{i=0..floor(n/2)} C(n+9-i, n-2i), n >= 0.
a(n) = a(n-1) + a(n-2) + C(n+8,8); n >= 0; a(-1)=0.
G.f.: 1/((x^2 + x - 1)*(x-1)^9). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

A137176 Hyperlucas number array T(r,n) = L(n)^(r), read by ascending antidiagonals (r >= 0, n >= 0).

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 4, 0, 1, 5, 8, 7, 0, 1, 6, 13, 15, 11, 0, 1, 7, 19, 28, 26, 18, 0, 1, 8, 26, 47, 54, 44, 29, 0, 1, 9, 34, 73, 101, 98, 73, 47, 0, 1, 10, 43, 107, 174, 199, 171, 120, 76, 0, 1, 11, 53, 150, 281, 373, 370, 291, 196, 123

Offset: 0

Jonathan Vos Post, Apr 04 2008

In Theorem 17, Dil and Mezo (2008) connect the hyperlucas numbers (this array) with the incomplete Lucas numbers (A324242). - Petros Hadjicostas, Sep 03 2019

			The array T(r,n) = L(n)^(r) begins:
.....|n=0|n=1|.n=2|.n=3|.n=4.|.n=5.|..n=6.|.n=7..|..n=8..|..n=9..|.n=10..|.in.OEIS
r=0..|.0.|.1.|..3.|..4.|...7.|..11.|...18.|...29.|....47.|....76.|...123.|.A000204
r=1..|.0.|.1.|..4.|..8.|..15.|..26.|...44.|...73.|...120.|...196.|...319.|.A027961
r=2..|.0.|.1.|..5.|.13.|..28.|..54.|...98.|..171.|...291.|...487.|...806.|.A023537
r=3..|.0.|.1.|..6.|.19.|..47.|.101.|..199.|..370.|...661.|..1148.|..1954.|.A027963
r=4..|.0.|.1.|..7.|.26.|..73.|.174.|..373.|..743.|..1404.|..2552.|..4506.|.A027964
r=5..|.0.|.1.|..8.|.34.|.107.|.281.|..654.|.1397.|..2801.|..5353.|..9859.|.A053298
r=6..|.0.|.1.|..9.|.43.|.150.|.431.|.1085.|.2482.|..5283.|.10636.|.20495.|.new
r=7..|.0.|.1.|.10.|.53.|.203.|.634.|.1719.|.4201.|..9484.|.20120.|.40615.|.new
r=8..|.0.|.1.|.11.|.64.|.267.|.901.|.2620.|.6821.|.16305.|.36425.|.77040.|.new
r=9..|.0.|.1.|.12.|.76.|.343.|1244.|.3864.|10685.|.26990.|.63415.|140455.|.new
For example, T(4,5) = L(5)^(4) = L(0)^(3) + L(1)^(3) + L(2)^(3) + L(3)^(3) + L(4)^(3) + L(5)^(3) = 0 + 1 + 6 + 19 + 47 + 101 = 174. - _Petros Hadjicostas_, Sep 03 2019

Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, arXiv:0803.4388 [math.NT], 2008.
Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, Applied Mathematics and Computation 206(2) (2008), 942-951.
Eric W. Weisstein, Steenrod Algebra.

Cf. A000204, A027961, A023537, A027963, A027964, A053298, A123736, A136431.

Cf. A038730, A038792, and A134511 for incomplete Fibonacci sequences, and A324242 for incomplete Lucas sequences.

L:= proc(r, n) option remember; `if`(n=0, 0, `if`(r=0,
      `if`(n<3, 2*n-1, L(0, n-2)+L(0, n-1)), L(r-1, n)+L(r, n-1)))
    end:
seq(seq(L(d-n, n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 03 2019

L[r_, n_] := L[r, n] = If[n == 0, 0, If[r == 0, If[n < 3, 2n-1, L[0, n-2] + L[0, n-1]], L[r-1, n] + L[r, n-1]]];
Table[L[d-n, n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)

T(r,n) = L(n)^(r) = Apply partial sum operator r times to Lucas numbers A000204.
From Petros Hadjicostas, Sep 03 2019: (Start)
T(r, n) = L(n)^(r) = Sum_{k = 0..n} L(k)^(r-1) for r >= 1, with T(0,n) = L(n)^(0) = L(n) = A000204(n), T(r,0) = L(0)^(r) = 0, and T(r,1) = L(1)^(r) = 1. (See Definition 13 in Dil and Mezo (2008).)
G.f. for row r: Sum_{n >= 0} L(n)^(r)*t^n = t * (1+2*t)/((1-t-t^2) * (1-t)^r). (Corrected from Proposition 14 in Dil and Mezo (2008).)
(End)

A123736 Triangle T(n,k) = Sum_{j=0..k/2} binomial(n-j-1,k-2*j), read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 7, 7, 5, 3, 2, 1, 1, 0, 1, 5, 11, 14, 12, 8, 5, 3, 2, 1, 1, 0, 1, 6, 16, 25, 26, 20, 13, 8, 5, 3, 2, 1, 1, 0, 1, 7, 22, 41, 51, 46, 33, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, 8, 29, 63, 92, 97, 79, 54, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1

Offset: 1

Roger L. Bagula, Nov 14 2006

Row sums give: A000225

			The triangle starts in row n=1 with columns 0 <= k < 2*n:
  1, 0;
  1, 1,  1,  0;
  1, 2,  2,  1,  1,  0;
  1, 3,  4,  3,  2,  1,  1,  0;
  1, 4,  7,  7,  5,  3,  2,  1,  1,  0;
  1, 5, 11, 14, 12,  8,  5,  3,  2,  1,  1, 0;
  1, 6, 16, 25, 26, 20, 13,  8,  5,  3,  2, 1, 1, 0;
  1, 7, 22, 41, 51, 46, 33, 21, 13,  8,  5, 3, 2, 1, 1, 0;
  1, 8, 29, 63, 92, 97, 79, 54, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0;

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Steenrod Algebra

Cf. A136431 (antidiagonals), A027926 (row-reversed), A004006 (column m=3)

Flat(List([1..10], n-> List([0..2*n-1], k-> Sum([0..Int(k/2)], j-> Binomial(n-j-1, k-2*j) )))); # G. C. Greubel, Sep 05 2019

[&+[Binomial(n-j-1, k-2*j): j in [0..Floor(k/2)]]: k in [0..2*n-1], n in [1..10]]; // G. C. Greubel, Sep 05 2019

seq(seq(sum(binomial(n-j-1, k-2*j), j=0..floor(k/2)), k=0..2*n-1), n=1..10); # G. C. Greubel, Sep 05 2019

Table[Sum[Binomial[n-j-1, k-2*j], {j,0,Floor[k/2]}], {n, 10}, {k, 0, 2*n-1}]//Flatten (* modified by G. C. Greubel, Sep 05 2019 *)

T(n,k) = sum(j=0, k\2, binomial(n-j-1, k-2*j));
for(n=1,10, for(k=0,2*n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 05 2019

[[sum(binomial(n-j-1, k-2*j) for j in (0..floor(k/2))) for k in (0..2*n-1)] for n in (1..10)] # G. C. Greubel, Sep 05 2019

A257838 Main diagonal of iterated partial sums array of Fibonacci numbers (starting with the first partial sums).

Original entry on oeis.org

0, 1, 4, 16, 63, 247, 967, 3785, 14820, 58060, 227612, 892926, 3505386, 13770404, 54129602, 212904952, 837885495, 3299264407, 12997784803, 51230474669, 202014314769, 796928589755, 3145066003589, 12416625685891, 49037912997003, 193734379979677, 765632076098287, 3026670770970925, 11968378998073935

Offset: 0

Luciano Ancora, May 10 2015

The array used here starts in row n=0 with the first partial sums of A000045. The array which starts with the Fibonacci numbers in row k=0 is shown in A136431. The diagonal of that array is given in A176085. - Wolfdieter Lang, Jun 03 2015

			This sequence is the main diagonal of the following array (see the comment and Example field of A136431):
0, 1, 2,  4,  7,  12, ...  A000071
0, 1, 3,  7, 14,  26, ...  A001924
0, 1, 4, 11, 25,  51, ...  A014162
0, 1, 5, 16, 41,  92, ...  A014166
0, 1, 6, 22, 63, 155, ...  A053739
0, 1, 7, 29, 92, 247, ...  A053295

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000045, A000071, A001924, A014162, A014166, A136431, A176085.

Table[DifferenceRoot[Function[{a, n},{(2*n + 4*n^2)*a[n] + (2 + 7*n + 15*n^2)*a[1 + n] + (8 - 6*n - 8*n^2)*a[2 + n] + (-2 + n + n^2)*a[3 + n] == 0, a[1] == 0, a[2] == 1, a[3] == 4, a[4] == 16}]][n], {n, 30}]

a(n):=sum(binomial(2*n-k,n-k)*fib(k),k,0,n); /* Vladimir Kruchinin, Oct 09 2016 */

x='x+O('x^50); concat([0], Vec(-(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x))) \\ G. C. Greubel, Apr 08 2017

a(n) = F^{n+1}(n), n >= 0, with the k-th iterated partial sum F^{k} of the Fibonacci number A000045. - Wolfdieter Lang, Jun 03 2015
Conjecture: n*(n-3)*a(n) +2*(-4*n^2+13*n-6)*a(n-1) +(15*n^2-53*n+48)*a(n-2) +2*(2*n-3)*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 10 2015
G.f.: -(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x). - Vladimir Kruchinin, Oct 09 2016
a(n) = Sum_{k=0..n} binomial(2*n-k,n-k)*F(k), where F(k) = A000045(k). - Vladimir Kruchinin, Oct 09 2016
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 09 2016

Name edited by Wolfdieter Lang, Jun 03 2015

cp's OEIS Frontend

A176085 a(n) = A136431(n,n).

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A136338 Primes in the array A136431 that are not Fibonacci numbers.

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A162741 Fibonacci-Pascal triangle; same as Pascal triangle, but beginning another Pascal triangle to the right of each row starting at row 2.

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A014166 Apply partial sum operator 4 times to Fibonacci numbers.

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A053296 Partial sums of A053295.

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A053308 Partial sums of A053296.

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