{"id":"n-8c08995d48487910","revision":0,"first_index":0,"fingerprint":"f6-9111d60e4a92b32c","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["784","1008","324"],["21952","42336","27216","5832"],["614656","1580544","1524096","653184","104976"],["17210368","55319040","71124480","45722880","14696640","1889568"],["481890304","1858719744","2987228160","2560481280","1234517760","317447424","34012224"],["13492928512","60718178304","117099343872","125463582720","80655160320","31109847552","6666395904","612220032"],["377801998336","1942981705728","4371708837888","5620768505856","4516688977920","2322868617216","746636341248","137137287168","11019960576"],["10578455953408","61203923730432","157381518163968","236072277245952","227641124487168","146340722884608","62717452664832","17279298183168","2777030065152","198359290368"],["296196766695424","1904122071613440","5508353135738880","9442891089838080","10623252476067840","8195080481538048","4390221686538240","1612734497095680","388784209121280","55540601303040","3570467226624"],["8293509467471872","58646959805693952","188508085089730560","363551306958766080","467423108946984960","420680798052286464","270437655890755584","124180556276367360","39915178803118080","8553252600668160","1099703905800192","64268410079232"],["232218265089212416","1791398044973924352","6333871659014946816","13572582126460600320","19631770575773368320","20192678306509750272","15144508729882312704","8344933381771886592","3352875019461918720","957964291274833920","184750256174432256","21594185786621952","1156831381426176"],["6502111422497947648","54339074030875705344","209593571261949149184","494041989403165851648","793996054397945118720","918766862946193637376","787514453953880260608","506259291827494453248","244089301416827682816","87174750506009886720","22416364415831113728","3930141813165195264","421086622839128064","20822964865671168"],["182059119829942534144","1638532078469482807296","6846723327890338873344","17605859986003728531456","31124645332399448653824","40017401141656433983488","38588208243740132769792","28350520342339689381888","15947167692566075277312","6834500439671175118848","2196803712751449145344","513538530253585514496","82532978076469100544","8162602227343097856","374813367582081024"],["5097655355238390956032","49155962354084484218880","221201830593380178984960","616205099510130498600960","1188395549055251675873280","1680730847949570227306496","1800783051374539529256960","1488402317972833692549120","956830061553964516638720","478415030776982258319360","184531511871121728208896","53921545676626479022080","11554616930705674076160","1714146467742050549760","157421614384474030080","6746640616477458432"],["142734349946674946768896","1468124742308656595337216","7078458578988165727518720","21235375736964497182556160","44366767164729395899269120","68451583625582496530300928","80675080701579370910711808","74089359827981054918000640","53582483447022012931768320","30618561969726864532439040","13778352886377089039597568","4831370492625732520378368","1294117096239035496529920","255979205849479548764160","35262441622122182737920","3022494996181901377536","121439531096594251776"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056","42420747482776576","1951354384207722496","89762301673555234816","4129065876983540801536","189937030341242876870656","8737103395697172336050176","401906756202069927458308096"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["784","1008","324"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["28","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195071,"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-541bf93e05058bc8"]},{"id":195072,"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-541bf93e05058bc8"]},{"id":195073,"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-541bf93e05058bc8"]}],"region_exact":{},"incoming_edges":[{"id":"e-541bf93e05058bc8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-8c08995d48487910","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":"6f970ab2cfd438e3b3404b5f3974cb7660a77ab14af87a1c76256d93bd0bd196","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-4b2df01c2184142e","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","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-42cfee53b6478a97","kind":"fingerprint_related","node_a_id":"n-4b2df01c2184142e","node_b_id":"n-8c08995d48487910","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-4b2df01c2184142e","other":{"id":"n-4b2df01c2184142e","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","1008","784"],["5832","27216","42336","21952"],["104976","653184","1524096","1580544","614656"]],"card_rows":[["324","1008","784"],["5832","27216","42336","21952"],["104976","653184","1524096","1580544","614656"],["1889568","14696640","45722880","71124480","55319040","17210368"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611"],"edges":["e-e460f89b751823ed"],"text":"A038321 \u2192 basis A013611"},"headline":"A038321 \u2192 basis A013611","chain":null}},{"id":"l-6b3a3c875bcda380","kind":"fingerprint_related","node_a_id":"n-1bb976829c17ec98","node_b_id":"n-8c08995d48487910","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-1bb976829c17ec98","other":{"id":"n-1bb976829c17ec98","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1008","1568"],["27216","84672","65856"],["653184","3048192","4741632","2458624"]],"card_rows":[["1008","1568"],["27216","84672","65856"],["653184","3048192","4741632","2458624"],["14696640","91445760","213373440","221276160","86051840"]],"row_sums_preview":["2576","177744","10901632","626843840","34601779968","1856962191616","97623155216384","5051998282447872"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013611","D"],"edges":["e-e460f89b751823ed","e-27550b5f4421b741"],"text":"A038321 \u2192 basis A013611 \u2192 D"},"headline":"A038321 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-7957ab7e5b0f6cfe","kind":"fingerprint_related","node_a_id":"n-6d0748781e5e30ed","node_b_id":"n-8c08995d48487910","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-6d0748781e5e30ed","other":{"id":"n-6d0748781e5e30ed","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","9"],["196","252","81"]],"card_rows":[["1"],["14","9"],["196","252","81"],["2744","5292","3402","729"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610","basis A013610"],"edges":["e-254f1b828fe6620e","e-08a03f724769d7c5"],"text":"A038303 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038303 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-d0c764bebb7469e7","kind":"fingerprint_related","node_a_id":"n-8c08995d48487910","node_b_id":"n-ce03950bfb372f88","fingerprint":"f6-9111d60e4a92b32c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9111d60e4a92b32c"},"other_id":"n-ce03950bfb372f88","other":{"id":"n-ce03950bfb372f88","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-9111d60e4a92b32c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","252","81"],["2744","5292","3402","729"],["38416","98784","95256","40824","6561"]],"card_rows":[["196","252","81"],["2744","5292","3402","729"],["38416","98784","95256","40824","6561"],["537824","1728720","2222640","1428840","459270","59049"]],"row_sums_preview":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-3a9169eda66e0655","e-066880bf2751c810"],"text":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038281 \u2192 x \u21a6 x+1 \u2192 basis A013610","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:              784,              1008,                324;\n  1:            21952,             42336,              27216,               5832;\n  2:           614656,           1580544,            1524096,             653184,             104976;\n  3:         17210368,          55319040,           71124480,           45722880,           14696640,            1889568;\n  4:        481890304,        1858719744,         2987228160,         2560481280,         1234517760,          317447424,           34012224;\n  5:      13492928512,       60718178304,       117099343872,       125463582720,        80655160320,        31109847552,         6666395904,          612220032;\n  6:     377801998336,     1942981705728,      4371708837888,      5620768505856,      4516688977920,      2322868617216,       746636341248,       137137287168,       11019960576;\n  7:   10578455953408,    61203923730432,    157381518163968,    236072277245952,    227641124487168,    146340722884608,     62717452664832,     17279298183168,     2777030065152,     198359290368;\n  8:  296196766695424,  1904122071613440,   5508353135738880,   9442891089838080,  10623252476067840,   8195080481538048,   4390221686538240,   1612734497095680,   388784209121280,   55540601303040,    3570467226624;\n  9: 8293509467471872, 58646959805693952, 188508085089730560, 363551306958766080, 467423108946984960, 420680798052286464, 270437655890755584, 124180556276367360, 39915178803118080, 8553252600668160, 1099703905800192, 64268410079232;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (28+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 1008*x + 324*x^2.","mathematica":"P[0, x_] := 784 + 1008*x + 324*x^2;\nP[n_, x_] := P[n, x] = Expand[(28 + 18*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 784 + 1008*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 18*x)*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-7dcf11eddf2b9a8a","steps":["basis A013611","reverse"],"edges":["e-e460f89b751823ed","e-541bf93e05058bc8"],"text":"A038321 \u2192 basis A013611 \u2192 reverse"},"headline":"A038321 \u2192 basis A013611 \u2192 reverse","chain":null}