{"id":"n-9f660947dd0578ca","revision":1,"first_index":0,"fingerprint":"f6-f1db139b52e8ecd3","name":"obreschkoff_combination of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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":true,"rows":[["2","1"],["2","5","3"],["2","14","30","18"],["2","30","154","306","180"],["2","55","554","2481","4680","2700"],["2","91","1589","13353","55053","99900","56700"],["2","140","3892","54856","413586","1609884","2835000","1587600"],["2","204","8484","185672","2299122","16094556","60013224","103194000","57153600"],["2","285","16908","542882","10261098","116287749","770654088","2778784380","4686595200","2571912000"],["2","385","31383","1415282","38699848","661287249","7020661563","44592409980","156503672100","259763112000","141455160000"],["2","506","54978","3363866","127698494","3123902118","49546855350","500108462334","3070153773000","10538081613000","17257529520000","9336040560000"],["2","650","91806","7409402","377907530","12719886582","286496244090","4289076270846","41575631051052","248187585141000","836141600934000","1353725881200000","728211163680000"],["2","819","147238","15312011","1021619742","45844264297","1410669753354","29798119514673","425925059703696","3993632376734232","23287560273396000","77220102599514000","123795897825600000","66267215894880000"],["2","1015","228137","29973203","2558542129","149186162957","6082139582835","174512850780897","3501095246177193","48174560724502584","439288266104298360","2510947573934058000","8213085213005610000","13054641531291360000","6958057668962400000"],["2","1240","343112","55996928","6001545004","445095680432","23449148660760","886858403887872","24053128792772418","462496974642838584","6163899667814514960","54888511485247531200","308388658393709460000","996778851792035760000","1572521033185502400000","834966920275488000000"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","10","64","672","10472","226688","6504960","238948864","10935054592","610036449792","40754715803648","3212430539816960","295021589700440064","31227876656033677312","3774195335661939392512"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"]],"recurrence":{"denominator":{"coeffs":[["8","0"],["0","2"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["8","4","0"],["0","10","2"],["0","6","5"],["0","0","4"],["0","0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162827,"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-4ea38c3ef421f3f7"]}],"region_exact":{},"incoming_edges":[{"id":"e-4ea38c3ef421f3f7","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-9f660947dd0578ca","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":"423525845232e861cd6cf80a3623f6a582d65a37b82c57deb5adc42bcb04aaf6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-ecd77b030b8fab94","role":"sequence","name":"Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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-ecd77b030b8fab94","role":"sequence","name":"Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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: 2,   1;\n  1: 2,   5,     3;\n  2: 2,  14,    30,      18;\n  3: 2,  30,   154,     306,      180;\n  4: 2,  55,   554,    2481,     4680,      2700;\n  5: 2,  91,  1589,   13353,    55053,     99900,      56700;\n  6: 2, 140,  3892,   54856,   413586,   1609884,    2835000,     1587600;\n  7: 2, 204,  8484,  185672,  2299122,  16094556,   60013224,   103194000,     57153600;\n  8: 2, 285, 16908,  542882, 10261098, 116287749,  770654088,  2778784380,   4686595200,   2571912000;\n  9: 2, 385, 31383, 1415282, 38699848, 661287249, 7020661563, 44592409980, 156503672100, 259763112000, 141455160000;\n","recurrence_sentence":"The row polynomials satisfy (8+(2*n^2+2*n)*x)*P(n,x) = (8+(6*n^2+10*n+4)*x+(n^4+4*n^3+5*n^2+2*n)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + (6*n^2 + 10*n + 4)*x + (n^4 + 4*n^3 + 5*n^2 + 2*n)*x^2)*P[n - 1, x])/(8 + (2*n^2 + 2*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (6*n^2 + 10*n + 4)*x + (n^4 + 4*n^3 + 5*n^2 + 2*n)*x^2)*v[i - 1])/(8 + (2*n^2 + 2*n)*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-ecd77b030b8fab94","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-4ea38c3ef421f3f7"],"text":"A128813 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A128813 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}