{"id":"n-373403bc1de177ac","revision":0,"first_index":0,"fingerprint":"f6-8987aa5000ed2f45","name":"basis_map_A013611 of Triangle T(n,k) read by rows, arising in enumeration of catafusenes.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["2","4"],["4","16","16"],["16","112","256","192"],["64","640","2368","3840","2304"],["256","3328","17152","43776","55296","27648"],["1024","16384","108544","380928","746496","774144","331776"],["4096","77824","630784","2826240","7557120","12054528","10616832","3981312"],["16384","360448","3457024","18874368","64143360","138903552","187121664","143327232","47775744"],["65536","1638400","18153472","116981760","483065856","1325334528","2415329280","2818768896","1911029760","573308928"],["262144","7340032","92274688","685768704","3336044544","11098128384","25565331456","40259026944","41469345792","25225592832","6879707136"],["1048576","32505856","457179136","3850371072","21573402624","84425048064","235438866432","467820085248","648985706496","598534520832","330225942528","82556485632"],["4194304","142606336","2218786816","20887633920","132498063360","596581023744","1954856042496","4696546738176","8209783848960","10181966561280","8503318020096","4292937252864","990677827584"],["16777216","620756992","10586423296","110175977472","780643860480","3976300855296","14978396454912","42244459462656","89197696253952","139245272432640","156196870815744","119211565252608","55477958344704","11888133931008"],["67108864","2684354560","49794777088","567740989440","4444687171584","25272929746944","107629196083200","348718595309568","863724298567680","1627353444777984","2295730752454656","2351208710799360","1652450616410112","713288035860480","142657607172096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","36","576","9216","147456","2359296","37748736","603979776","9663676416","154618822656","2473901162496","39582418599936","633318697598976","10133099161583616"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"],["4","16","16"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-8","-24"],["4","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204785,"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-baa2790cd8bee875"]},{"id":204786,"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-baa2790cd8bee875"]},{"id":204787,"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-baa2790cd8bee875"]},{"id":204788,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-baa2790cd8bee875"]}],"region_exact":{},"incoming_edges":[{"id":"e-baa2790cd8bee875","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-373403bc1de177ac","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"1e749a9d4c30f97c3984e43afcac89599ef8703c6ff4a7b771940206132a6307","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-df53153327eff8f9","role":"sequence","name":"Triangle T(n,k) read by rows, arising in enumeration of catafusenes.","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-aff693e7b2660ece","kind":"fingerprint_related","node_a_id":"n-b5ecca697c4721f9","node_b_id":"n-373403bc1de177ac","fingerprint":"f6-8987aa5000ed2f45","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8987aa5000ed2f45"},"other_id":"n-b5ecca697c4721f9","other":{"id":"n-b5ecca697c4721f9","name":"shift_x_plus_one of shift_x_plus_one of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-8987aa5000ed2f45","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","4","1"],["12","16","7","1"],["36","60","37","10","1"]],"card_rows":[["4","4","1"],["12","16","7","1"],["36","60","37","10","1"],["108","216","171","67","13","1"]],"row_sums_preview":["9","36","144","576","2304","9216","36864","147456"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-34caf7309ed0b453","e-50a010689dfb3262"],"text":"A159854 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A159854 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","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,       4;\n  2:     4,      16,       16;\n  3:    16,     112,      256,       192;\n  4:    64,     640,     2368,      3840,      2304;\n  5:   256,    3328,    17152,     43776,     55296,      27648;\n  6:  1024,   16384,   108544,    380928,    746496,     774144,     331776;\n  7:  4096,   77824,   630784,   2826240,   7557120,   12054528,   10616832,    3981312;\n  8: 16384,  360448,  3457024,  18874368,  64143360,  138903552,  187121664,  143327232,   47775744;\n  9: 65536, 1638400, 18153472, 116981760, 483065856, 1325334528, 2415329280, 2818768896, 1911029760, 573308928;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(12*n-24)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 4 + 16*x + 16*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 4 + 16*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (12*n - 24)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 4 + 16*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (12*n - 24)*x)*v[i - 1])/(n - 2)); 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-df53153327eff8f9","steps":["basis A013611"],"edges":["e-baa2790cd8bee875"],"text":"A024462 \u2192 basis A013611"},"headline":"A024462 \u2192 basis A013611","chain":null}
