{"id":"n-806f02861480e04e","revision":1,"first_index":1,"fingerprint":"f6-b3b7fb695f9c1be8","name":"Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["3","4","1"],["11","19","9","1"],["49","104","70","16","1"],["257","641","550","190","25","1"],["1539","4380","4531","2080","425","36","1"],["10299","32803","39515","22491","6265","833","49","1"],["75905","266768","365324","247072","87206","16016","1484","64","1"],["609441","2337505","3575820","2792476","1192086","281190","36204","2460","81","1"],["5284451","21925236","36971461","32659840","16333150","4719624","786282","74400","3855","100","1"],["49134923","218946003","402741581","396255541","226854430","77684398","15982890","1963962","141735","5775","121","1"],["487026929","2316939256","4610187154","4991365808","3214789391","1273177488","312957436","47710080","4477407","253880","8338","144","1"],["5120905441","25878593313","55316069874","65268062938","46656245207","20975230991","6009565276","1100247148","128395839","9467887","432146","11674","169","1"],["56878092067","304020964876","694067320311","885442472096","694926579711","349498569420","114503039651","24567605920","3453398377","316928612","18799781","704704","15925","196","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","40","240","1664","12992","112256","1059840","10827264","118758400","1389711360","17258893312","226463227904","3127694491648"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A154602","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160375,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-18b4793e32bd250f"]},{"id":160376,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-18b4793e32bd250f"]},{"id":160377,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-18b4793e32bd250f"]}],"region_exact":{},"incoming_edges":[{"id":"e-18b4793e32bd250f","operation":{"map":"A154602","operation":"basis_map"},"operation_detail":"A154602","conclusion_id":"n-806f02861480e04e","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7be2569ce793b7954765a6d17bbcbc74aadd6c170a5f9721f9d37028c18e6067","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-61ad677adce5c0b7","kind":"oeis_match","node_a_id":"n-806f02861480e04e","node_b_id":null,"fingerprint":null,"oeis_id":"A055882","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A055882","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-709ce6384d021709","kind":"oeis_match","node_a_id":"n-806f02861480e04e","node_b_id":null,"fingerprint":null,"oeis_id":"A154602","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A154602","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A154602","relation":"identification"},{"oeis_id":"A055882","relation":"match_row_sums"},{"oeis_id":"A154602","relation":"match_triangle"}],"bfiles":[{"node_id":"n-806f02861480e04e","oeis_id":"A055882","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":31259},{"node_id":"n-806f02861480e04e","oeis_id":"A154602","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":27455}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,       1;\n   3:      3,       4,       1;\n   4:     11,      19,       9,       1;\n   5:     49,     104,      70,      16,       1;\n   6:    257,     641,     550,     190,      25,      1;\n   7:   1539,    4380,    4531,    2080,     425,     36,     1;\n   8:  10299,   32803,   39515,   22491,    6265,    833,    49,    1;\n   9:  75905,  266768,  365324,  247072,   87206,  16016,  1484,   64,  1;\n  10: 609441, 2337505, 3575820, 2792476, 1192086, 281190, 36204, 2460, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + (2 + 2*x)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A154602","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      1,       1;\n  2:      3,       4,       1;\n  3:     11,      19,       9,       1;\n  4:     49,     104,      70,      16,       1;\n  5:    257,     641,     550,     190,      25,      1;\n  6:   1539,    4380,    4531,    2080,     425,     36,     1;\n  7:  10299,   32803,   39515,   22491,    6265,    833,    49,    1;\n  8:  75905,  266768,  365324,  247072,   87206,  16016,  1484,   64,  1;\n  9: 609441, 2337505, 3575820, 2792476, 1192086, 281190, 36204, 2460, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (1 + x)*v[i - 1] + (2 + 2*x)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A154602"],"edges":["e-18b4793e32bd250f"],"text":"basis-map coefficients of A026729 \u2192 basis A154602"},"headline":"A154602: Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].","chain":"basis-map coefficients of A026729 \u2192 basis A154602"}
