{"id":"n-4b2633a9a6081341","revision":1,"first_index":0,"fingerprint":"f6-6d9e12a32349f2fa","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","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":[["3","2"],["6","18","4"],["12","104","92","8"],["24","504","1144","424","16"],["48","2240","10720","10080","1840","32"],["96","9504","85888","164288","78048","7712","64"],["192","39296","625728","2153856","2075456","557184","31680","128"],["384","160128","4289152","24587136","41983872","23137408","3771264","128640","256"],["768","647168","28233728","255646208","710885888","693489152","236797952","24622592","518912","512"],["1536","2603520","180794368","2489430016","10642729984","16772531200","10204887040","2280865792","156818944","2085376","1024"],["3072","10446848","1135799296","23120750592","145792063488","348926791680","342796990464","138076606464","21020679168","981583872","8363008","2048"],["6144","41859072","7040456704","207369390080","1869581402112","6499850858496","9651929690112","6302845931520","1754625546240","187484510208","6069319680","33499136","4096"],["12288","167591936","43230633984","1811886170112","22801473064960","111379035070464","238458979074048","235306900242432","106945770541056","21254216908800","1631442116608","37203173376","134098944","8192"],["24576","670703616","263674200064","15520502661120","267503216369664","1789110359810048","5330914286911488","7561615860006912","5206986421739520","1704966062333952","248060885630976","13932291178496","226638618624","536616960","16384"],["49152","2683535360","1600556711936","130945211138048","3044273465245696","27312259654057984","110204583194214400","216484031175426048","214253329602330624","106786707132284928","25874796808093696","2811733082275840","117276594159616","1374588796928","2146942976","32768"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","28","216","2112","24960","345600","5483520","98058240","1950842880","42732748800","1021870080000","26486872473600","739670438707200","22139101406822400","707022915895296000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3","2"],["6","18","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-2"],["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162811,"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-1d58a9b88feb4822"]}],"region_exact":{},"incoming_edges":[{"id":"e-1d58a9b88feb4822","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-4b2633a9a6081341","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":"a03110ae2bc579460139e8d3dd954da0fcca0b4e58aad5e93e38baba32eb2972","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-41809911e05bb4ba","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","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-41809911e05bb4ba","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","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:    3,       2;\n  1:    6,      18,         4;\n  2:   12,     104,        92,          8;\n  3:   24,     504,      1144,        424,          16;\n  4:   48,    2240,     10720,      10080,        1840,          32;\n  5:   96,    9504,     85888,     164288,       78048,        7712,          64;\n  6:  192,   39296,    625728,    2153856,     2075456,      557184,       31680,        128;\n  7:  384,  160128,   4289152,   24587136,    41983872,    23137408,     3771264,     128640,       256;\n  8:  768,  647168,  28233728,  255646208,   710885888,   693489152,   236797952,   24622592,    518912,     512;\n  9: 1536, 2603520, 180794368, 2489430016, 10642729984, 16772531200, 10204887040, 2280865792, 156818944, 2085376, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(2*n-2)*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + (1+(2*n+2)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 18*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 18*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (2*n - 2)*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x)*P[n - 1, x] + (2*x - 2*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] = 3 + 2*x; v[2] = 6 + 18*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (2*n - 2)*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (1 + (2*n + 2)*x)*v[i - 1] + (2*x - 2*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-41809911e05bb4ba","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-1d58a9b88feb4822"],"text":"A257609 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257609 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}