{"id":"n-a375f9a732b31b5d","revision":1,"first_index":0,"fingerprint":"f6-0200050f881fc484","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) = 3*x + 5.","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":[["6","5"],["30","85","25"],["150","995","940","125"],["750","9985","21045","9195","625"],["3750","92255","368540","357480","83810","3125"],["18750","811165","5595065","10103890","5329840","731105","15625"],["93750","6911195","77519460","235525335","231016510","73000785","6198840","78125"],["468750","57680185","1009487605","4832003925","7901944925","4678506875","943398015","51575095","390625"],["2343750","474800855","12585498820","90649451450","230295832660","227225891660","86980432180","11699780990","423694510","1953125"],["11718750","3872234965","152055220905","1593503645220","5986147836360","9127405961190","5849571396150","1519108711860","140830240770","3450884205","9765625"],["58593750","31382176595","1795249037180","26683707587265","142937478341700","319561660487310","316420402362480","138681410590770","25316654096310","1658291031375","27943011140","48828125"],["292968750","253254678385","20835768796565","430598654600815","3199242270556065","10090317062094090","14600873086634370","9916834138073310","3087244567552560","407077249541685","19208859941785","225370260995","244140625"],["1464843750","2037902661455","238733334804300","6752668545990300","68090746805200910","294172845237518895","597236353984485720","592725797959945440","287455253370001650","65434100892249165","6366427065245180","219777729540900","1812825369210","1220703125"],["7324218750","16366941994765","2708944727668345","103551683586338790","1392669827575751800","8053174597555627615","22238109672967488375","30896145301917858180","21939556720245061830","7833810090990826875","1333988395229492935","97423989855897910","2491204900788900","14555581469305","6103515625"],["36621093750","131276112129995","30513099708435700","1560131646818821115","27591309306653007890","209611849880809441975","768324930044883490760","1446456119952605370375","1437798084323655000570","754438547922194056545","203170709617394366900","26360872595009853745","1465416601690591550","28038261749641565","116727854879440","30517578125"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["11","140","2210","41600","908960","22605440","630344000","19475456000","660339680000","24375967616000","973114286144000","41772710819840000","1918696558338560000","93893661365504000000","4876836771324277760000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["6","5"],["30","85","25"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","-1"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","5"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162870,"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-148e9e46774006f0"]}],"region_exact":{},"incoming_edges":[{"id":"e-148e9e46774006f0","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-a375f9a732b31b5d","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":"770e3b84dacc4c5ad73bc78a2d1c32d64b53c3be3ef955c375a87255bda7578d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-f3a4ae052af3da90","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) = 3*x + 5.","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-f3a4ae052af3da90","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) = 3*x + 5.","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:        6,          5;\n  1:       30,         85,           25;\n  2:      150,        995,          940,           125;\n  3:      750,       9985,        21045,          9195,           625;\n  4:     3750,      92255,       368540,        357480,         83810,          3125;\n  5:    18750,     811165,      5595065,      10103890,       5329840,        731105,         15625;\n  6:    93750,    6911195,     77519460,     235525335,     231016510,      73000785,       6198840,         78125;\n  7:   468750,   57680185,   1009487605,    4832003925,    7901944925,    4678506875,     943398015,      51575095,       390625;\n  8:  2343750,  474800855,  12585498820,   90649451450,  230295832660,  227225891660,   86980432180,   11699780990,    423694510,    1953125;\n  9: 11718750, 3872234965, 152055220905, 1593503645220, 5986147836360, 9127405961190, 5849571396150, 1519108711860, 140830240770, 3450884205, 9765625;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(3*n-1)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (4+(3*n+5)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 6 + 5*x, P(1,x) = 30 + 85*x + 25*x^2.","mathematica":"P[0, x_] := 6 + 5*x;\nP[1, x_] := 30 + 85*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(5 + (3*n - 1)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (4 + (3*n + 5)*x)*P[n - 1, x] + (3*x - 3*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] = 6 + 5*x; v[2] = 30 + 85*x + 25*x^2; for(n = 2, N, my(i = n + 1); v[i] = (5 + (3*n - 1)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (4 + (3*n + 5)*x)*v[i - 1] + (3*x - 3*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-f3a4ae052af3da90","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-148e9e46774006f0"],"text":"A257624 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257624 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}