{"id":"n-2a4db0762a391b72","revision":0,"first_index":0,"fingerprint":"f6-28d339a36b0d6fb3","name":"basis_map_A013611 of basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","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"],["4","12"],["28","132","144"],["280","1656","3024","1728"],["3640","24888","59184","58752","20736"],["58240","441888","1245600","1650240","1036800","248832"],["1106560","9094752","28969056","46301760","39502080","17169408","2985984"],["24344320","213363264","746456256","1366267392","1424666880","851751936","271724544","35831808"],["608608000","5626213440","21221765568","43114159872","52011880704","38389800960","17014136832","4156489728","429981696"],["17041024000","164837272320","661723997184","1461857663232","1973702578176","1699056995328","937073442816","320551354368","61917364224","5159780352"],["528271744000","5314447729920","22491491180544","53258275526400","78727071882240","76355197793280","49437960671232","21181973299200","5766054543360","902961561600","61917364224"],["17961239296000","187030483745280","828484072897536","2080679262064128","3315819750312960","3540801587558400","2597153036341248","1313442620227584","450229534064640","99893347614720","12940729122816","743008370688"],["664565853952000","7135662770127360","32898276502152192","86926941571143168","147653481906349056","170799495743416320","138584281395326976","79763213384515584","32419804203122688","9098808270520320","1677527148920832","182780059189248","8916100448256"],["26582634158080000","293401301052518400","1401559013327616000","3871856980871553024","6949262575107680256","8603821612612841472","7592965204734074880","4853539912124547072","2253950728739094528","752989981258285056","176286785203077120","27441528154619904","2550004728201216","106993205379072"],["1143053268797440000","12935247555155251200","63787853185717708800","183308558337408172032","345280574500088887296","453355480243644346368","429743363154919317504","299817798678164422656","155162360281275629568","59425977938975391744","16616211538831736832","3295427133085581312","438948541168091136","35200764569714688","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","304","6688","167200","4681600","145129600","4934406400","182573036800","7302921472000","314025623296000","14445178671616000","707813754909184000","36806315255277568000","2024347339040266240000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","12"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":206188,"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-09500223adf3fe9c"]},{"id":206189,"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-09500223adf3fe9c"]},{"id":206190,"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-09500223adf3fe9c"]},{"id":206191,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-09500223adf3fe9c"]}],"region_exact":{},"incoming_edges":[{"id":"e-09500223adf3fe9c","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-2a4db0762a391b72","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":"06170d6738e791ca3e0c2181c5a76cf62c21f810746d05f2e5272b2171092601","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b0592fccca597966","role":"sequence","name":"basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x ->  x + 1","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":[],"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:           4,           12;\n  2:          28,          132,          144;\n  3:         280,         1656,         3024,          1728;\n  4:        3640,        24888,        59184,         58752,         20736;\n  5:       58240,       441888,      1245600,       1650240,       1036800,        248832;\n  6:     1106560,      9094752,     28969056,      46301760,      39502080,      17169408,      2985984;\n  7:    24344320,    213363264,    746456256,    1366267392,    1424666880,     851751936,    271724544,     35831808;\n  8:   608608000,   5626213440,  21221765568,   43114159872,   52011880704,   38389800960,  17014136832,   4156489728,   429981696;\n  9: 17041024000, 164837272320, 661723997184, 1461857663232, 1973702578176, 1699056995328, 937073442816, 320551354368, 61917364224, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+1+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n + 1 + 12*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 1 + 12*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-1a0f876fa1441737","steps":["basis A013610","basis A013611"],"edges":["e-6e846af4747cc041","e-09500223adf3fe9c"],"text":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 basis A013611"},"headline":"basis-map coefficients of A371080, shifted \u2192 basis A013610 \u2192 basis A013611","chain":null}