{"id":"n-b8bb389ae5897924","revision":1,"first_index":0,"fingerprint":"f6-57560575826fed2a","name":"obreschkoff_combination of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*n + 3.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4","3"],["12","57","9"],["36","675","630","27"],["108","6777","19251","6183","81"],["324","63747","427356","414882","57672","243"],["972","583929","8094573","17575218","7743438","526581","729"],["2916","5291811","140342706","585180693","575700048","133183197","4766202","2187"],["8748","47753145","2308363191","16873988769","31240781055","16438631427","2180471805","42989859","6561"],["26244","430211331","36745746552","443583987444","1396934632656","1382567840370","429453719208","34613027892","387230220","19683"],["78732","3873358521","572869050129","10955768093172","54919174744620","91539545008734","53980190891694","10563060120516","538686233568","3486154545","59049"],["236196","34865068707","8811493845678","259080031271871","1973263614220224","5154455229133590","5117834858869476","1930239191421702","249089840201484","8276680970631","31378992894","177147"],["708588","313801561593","134347990204539","5939706011853483","66428028238329693","258572567724045642","400078273733261982","255427996479396390","64756869078179862","5699427665497677","126106174302615","282422804895","531441"],["2125764","2824266135555","2036683887214932","133148110462794486","2130628695863400792","11909205910840272093","27208432531503327240","27066926825494110900","11720205942599205396","2071811532797074413","127583030787501348","1910891514473910","2541844039248","1594323"],["6377292","25418564218233","30760383732624117","2935885059164301894","65881158820643902554","514062580816720894491","1666977320466815309199","2440021087114161088356","1651819599552161470392","504473003569985089887","63942495398309622627","2810338712264015478","28852316484285762","22876722304749","4782969"],["19131876","228767623222563","463449408629991786","63949151367846071577","1980602615503769404176","21102181450382396904075","94243748419008514742094","194655911752706533409673","193882867986059187630516","93105991400273074166193","20663589605633055716742","1919596005164624083443","61169573039314680216","434622510709924257","205890907295106","14348907"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["7","78","1368","32400","964224","34525440","1444469760","69132994560","3724716441600","223076653793280","14701117184409600","1057163919320678400","82371895521863270400","6912983130505740288000","621680506465716220723200"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","3"],["12","57","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","-9"],["0","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"],["0","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162878,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-f5de3c16d688990e"]}],"region_exact":{},"incoming_edges":[{"id":"e-f5de3c16d688990e","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-b8bb389ae5897924","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"ddf0149e9b6868368d17da33d0be3480f72f000e6ee0f83b455d9fb35afa7b0c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-444025a5f2f6e0e5","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*n + 3.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-444025a5f2f6e0e5","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*n + 3.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     4,          3;\n  1:    12,         57,            9;\n  2:    36,        675,          630,             27;\n  3:   108,       6777,        19251,           6183,             81;\n  4:   324,      63747,       427356,         414882,          57672,            243;\n  5:   972,     583929,      8094573,       17575218,        7743438,         526581,            729;\n  6:  2916,    5291811,    140342706,      585180693,      575700048,      133183197,        4766202,           2187;\n  7:  8748,   47753145,   2308363191,    16873988769,    31240781055,    16438631427,     2180471805,       42989859,         6561;\n  8: 26244,  430211331,  36745746552,   443583987444,  1396934632656,  1382567840370,   429453719208,    34613027892,    387230220,      19683;\n  9: 78732, 3873358521, 572869050129, 10955768093172, 54919174744620, 91539545008734, 53980190891694, 10563060120516, 538686233568, 3486154545, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(6*n-9)*x)*P(n-2,x) + (6*x-6*x^2)*d/dx P(n-2,x) + (2+(6*n+3)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 57*x + 9*x^2.","mathematica":"P[0, x_] := 4 + 3*x;\nP[1, x_] := 12 + 57*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + (6*n - 9)*x)*P[n - 2, x] + (6*x - 6*x^2)*D[P[n - 2, x], x] + (2 + (6*n + 3)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 3*x; v[2] = 12 + 57*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (6*n - 9)*x)*v[i - 2] + (6*x - 6*x^2)*deriv(v[i - 2], x) + (2 + (6*n + 3)*x)*v[i - 1] + (6*x - 6*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-444025a5f2f6e0e5","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-f5de3c16d688990e"],"text":"A257625 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257625 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}