{"id":"n-22db44d78c07f843","revision":0,"first_index":0,"fingerprint":"f6-99cef2a9f1688781","name":"reversal of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["2","3","1"],["4","12","13","6","1"],["8","36","66","63","33","9","1"],["16","96","248","360","321","180","62","12","1"],["32","240","800","1560","1970","1683","985","390","100","15","1"],["64","576","2352","5760","9420","10836","8989","5418","2355","720","147","18","1"],["128","1344","6496","19152","38472","55692","59906","48639","29953","13923","4809","1197","203","21","1"],["256","3072","17152","59136","140896","245952","325360","332688","265729","166344","81340","30744","8806","1848","268","24","1"],["512","6912","43776","172800","476352","973728","1529472","1887408","1854882","1462563","927441","471852","191184","60858","14886","2700","342","27","1"],["1024","15360","108800","483840","1514880","3549312","6456480","9336960","10901460","10377180","8097453","5188590","2725365","1167120","403530","110916","23670","3780","425","30","1"],["2048","33792","264704","1309440","4590080","12127104","25075776","41592672","56270280","62795700","58227906","45046719","29113953","15698925","7033785","2599542","783618","189486","35860","5115","517","33","1"],["4096","73728","632832","3446784","13373184","39333888","91122944","170539776","262394352","335994912","361113192","327572856","251595969","163786428","90278298","41999364","16399647","5329368","1423796","307296","52239","6732","618","36","1"],["8192","159744","1490944","8865792","37719552","122234112","313620736","653782272","1127530976","1629712656","1992605472","2074480200","1847023698","1409933619","923511849","518620050","249075684","101857041","35235343","10215348","2450162","477477","73671","8658","728","39","1"],["16384","344064","3469312","22364160","103527424","366492672","1031663360","2370660864","4530029504","7295800512","10001879888","11756489472","11910093468","10435418532","7923848253","5217709266","2977523367","1469561184","625117493","227993766","70781711","18520788","4029935","715806","101101","10920","847","42","1"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","6","36","216","1296","7776","46656","279936","1679616","10077696","60466176","362797056","2176782336","13060694016","78364164096"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179678,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f6136b73603bfaf7"]},{"id":179679,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-f6136b73603bfaf7"]},{"id":179680,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-f6136b73603bfaf7"]}],"region_exact":{},"incoming_edges":[{"id":"e-f6136b73603bfaf7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":2}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-22db44d78c07f843","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"9b960466d44e79092471398691da368e954a402fa7a71909447af56540d4b184","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eed4f744ce063213","role":"sequence","name":"Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a5c5b4030d5459e7","kind":"fingerprint_related","node_a_id":"n-eed4f744ce063213","node_b_id":"n-22db44d78c07f843","fingerprint":"f6-99cef2a9f1688781","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-99cef2a9f1688781"},"other_id":"n-eed4f744ce063213","other":{"id":"n-eed4f744ce063213","name":"Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","first_index":0,"fingerprint":"f6-99cef2a9f1688781","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3","2"],["1","6","13","12","4"]],"card_rows":[["1"],["1","3","2"],["1","6","13","12","4"],["1","9","33","63","66","36","8"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A200536"],"derivation":{"root":"n-eed4f744ce063213","steps":[],"edges":[],"text":"A200536"},"headline":"A200536: Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","chain":null}}],"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:   1;\n  1:   2,    3,     1;\n  2:   4,   12,    13,      6,      1;\n  3:   8,   36,    66,     63,     33,      9,       1;\n  4:  16,   96,   248,    360,    321,    180,      62,      12,       1;\n  5:  32,  240,   800,   1560,   1970,   1683,     985,     390,     100,      15,      1;\n  6:  64,  576,  2352,   5760,   9420,  10836,    8989,    5418,    2355,     720,    147,     18,      1;\n  7: 128, 1344,  6496,  19152,  38472,  55692,   59906,   48639,   29953,   13923,   4809,   1197,    203,    21,     1;\n  8: 256, 3072, 17152,  59136, 140896, 245952,  325360,  332688,  265729,  166344,  81340,  30744,   8806,  1848,   268,   24,   1;\n  9: 512, 6912, 43776, 172800, 476352, 973728, 1529472, 1887408, 1854882, 1462563, 927441, 471852, 191184, 60858, 14886, 2700, 342, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*x + x^2)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*x + x^2)*v[i - 1]); 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-eed4f744ce063213","steps":["reverse"],"edges":["e-f6136b73603bfaf7"],"text":"A200536 \u2192 reverse"},"headline":"A200536 \u2192 reverse","chain":null}
