{"id":"n-4b2df01c2184142e","revision":1,"first_index":0,"fingerprint":"f6-9111d60e4a92b32c","name":"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":[["324","1008","784"],["5832","27216","42336","21952"],["104976","653184","1524096","1580544","614656"],["1889568","14696640","45722880","71124480","55319040","17210368"],["34012224","317447424","1234517760","2560481280","2987228160","1858719744","481890304"],["612220032","6666395904","31109847552","80655160320","125463582720","117099343872","60718178304","13492928512"],["11019960576","137137287168","746636341248","2322868617216","4516688977920","5620768505856","4371708837888","1942981705728","377801998336"],["198359290368","2777030065152","17279298183168","62717452664832","146340722884608","227641124487168","236072277245952","157381518163968","61203923730432","10578455953408"],["3570467226624","55540601303040","388784209121280","1612734497095680","4390221686538240","8195080481538048","10623252476067840","9442891089838080","5508353135738880","1904122071613440","296196766695424"],["64268410079232","1099703905800192","8553252600668160","39915178803118080","124180556276367360","270437655890755584","420680798052286464","467423108946984960","363551306958766080","188508085089730560","58646959805693952","8293509467471872"],["1156831381426176","21594185786621952","184750256174432256","957964291274833920","3352875019461918720","8344933381771886592","15144508729882312704","20192678306509750272","19631770575773368320","13572582126460600320","6333871659014946816","1791398044973924352","232218265089212416"],["20822964865671168","421086622839128064","3930141813165195264","22416364415831113728","87174750506009886720","244089301416827682816","506259291827494453248","787514453953880260608","918766862946193637376","793996054397945118720","494041989403165851648","209593571261949149184","54339074030875705344","6502111422497947648"],["374813367582081024","8162602227343097856","82532978076469100544","513538530253585514496","2196803712751449145344","6834500439671175118848","15947167692566075277312","28350520342339689381888","38588208243740132769792","40017401141656433983488","31124645332399448653824","17605859986003728531456","6846723327890338873344","1638532078469482807296","182059119829942534144"],["6746640616477458432","157421614384474030080","1714146467742050549760","11554616930705674076160","53921545676626479022080","184531511871121728208896","478415030776982258319360","956830061553964516638720","1488402317972833692549120","1800783051374539529256960","1680730847949570227306496","1188395549055251675873280","616205099510130498600960","221201830593380178984960","49155962354084484218880","5097655355238390956032"],["121439531096594251776","3022494996181901377536","35262441622122182737920","255979205849479548764160","1294117096239035496529920","4831370492625732520378368","13778352886377089039597568","30618561969726864532439040","53582483447022012931768320","74089359827981054918000640","80675080701579370910711808","68451583625582496530300928","44366767164729395899269120","21235375736964497182556160","7078458578988165727518720","1468124742308656595337216","142734349946674946768896"]],"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:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["324","1008","784"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["18","28"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168270,"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-e460f89b751823ed"]},{"id":168271,"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-e460f89b751823ed"]},{"id":168272,"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-e460f89b751823ed"]}],"region_exact":{},"incoming_edges":[{"id":"e-e460f89b751823ed","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-4b2df01c2184142e","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":"c0b3c345481ec8a243389fcb31d8bb4c6027b8429d19918ff843f5d08a14c30a","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-7dcf11eddf2b9a8a","role":"sequence","name":"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-7b60090f3b035bfd","kind":"oeis_match","node_a_id":"n-4b2df01c2184142e","node_b_id":null,"fingerprint":null,"oeis_id":"A009990","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009990","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009990","relation":"match_row_sums"}],"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:            324,             1008,              784;\n  1:           5832,            27216,            42336,             21952;\n  2:         104976,           653184,          1524096,           1580544,             614656;\n  3:        1889568,         14696640,         45722880,          71124480,           55319040,           17210368;\n  4:       34012224,        317447424,       1234517760,        2560481280,         2987228160,         1858719744,          481890304;\n  5:      612220032,       6666395904,      31109847552,       80655160320,       125463582720,       117099343872,        60718178304,        13492928512;\n  6:    11019960576,     137137287168,     746636341248,     2322868617216,      4516688977920,      5620768505856,      4371708837888,      1942981705728,       377801998336;\n  7:   198359290368,    2777030065152,   17279298183168,    62717452664832,    146340722884608,    227641124487168,    236072277245952,    157381518163968,     61203923730432,     10578455953408;\n  8:  3570467226624,   55540601303040,  388784209121280,  1612734497095680,   4390221686538240,   8195080481538048,  10623252476067840,   9442891089838080,   5508353135738880,   1904122071613440,   296196766695424;\n  9: 64268410079232, 1099703905800192, 8553252600668160, 39915178803118080, 124180556276367360, 270437655890755584, 420680798052286464, 467423108946984960, 363551306958766080, 188508085089730560, 58646959805693952, 8293509467471872;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+28*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1008*x + 784*x^2.","mathematica":"P[0, x_] := 324 + 1008*x + 784*x^2;\nP[n_, x_] := P[n, x] = Expand[(18 + 28*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] = 324 + 1008*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 28*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"],"edges":["e-e460f89b751823ed"],"text":"A038321 \u2192 basis A013611"},"headline":"A038321 \u2192 basis A013611","chain":null}