{"id":"n-c47b645304a2946e","revision":1,"first_index":0,"fingerprint":"f6-44d4fa66c7d58b2b","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","4"],["16","40","16"],["64","296","296","64"],["256","1928","3552","1928","256"],["1024","11688","34808","34808","11688","1024"],["4096","67656","302352","487312","302352","67656","4096"],["16384","379240","2423016","5830000","5830000","2423016","379240","16384"],["65536","2076424","18330496","62617144","93280000","62617144","18330496","2076424","65536"],["262144","11168552","132823640","621624968","1309794296","1309794296","621624968","132823640","11168552","262144"],["1048576","59250632","930964464","5812434816","16694604048","23576297328","16694604048","5812434816","930964464","59250632","1048576"],["4194304","310933224","6356045000","51858617280","197493615360","379132716432","379132716432","197493615360","51858617280","6356045000","310933224","4194304"],["16777216","1617580680","42489335136","445638905960","2202252330240","5584624216848","7582654328640","5584624216848","2202252330240","445638905960","42489335136","1617580680","16777216"],["67108864","8356338856","279199721016","3714323033624","23411324419400","76688645914512","137257409671728","137257409671728","76688645914512","23411324419400","3714323033624","279199721016","8356338856","67108864"],["268435456","42922544968","1808899747792","30188257050608","239291117825936","994545030682808","2292837847691424","3019663012778016","2292837847691424","994545030682808","239291117825936","30188257050608","1808899747792","42922544968","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","72","720","7920","95040","1235520","17297280","259459200","4151347200","70572902400","1270312243200","24135932620800","482718652416000","10137091700736000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A257606","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","3"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159719,"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":false,"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-de9fa36753ad8047"]},{"id":159720,"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-de9fa36753ad8047"]},{"id":159721,"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-de9fa36753ad8047"]},{"id":159722,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-de9fa36753ad8047"]}],"region_exact":{},"incoming_edges":[{"id":"e-de9fa36753ad8047","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-c47b645304a2946e","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"d932f8515b38917cf0185e4950d729c8df0d6e2d4930e6f497c2d54fffa9df0b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-9a33444ab708fa7d","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-91ed3144de9cf292","conclusion_name":"gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4."},{"id":"e-7b496eddb4982c99","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-8ae9deae4758feb6","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4."},{"id":"e-7b496eddb4982c99","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-8ae9deae4758feb6","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4."}],"used_by_total":3,"links":[{"id":"l-147af44937864303","kind":"oeis_match","node_a_id":"n-c47b645304a2946e","node_b_id":null,"fingerprint":null,"oeis_id":"A257606","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A257606","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2777089c0954b0ba","kind":"oeis_match","node_a_id":"n-c47b645304a2946e","node_b_id":null,"fingerprint":null,"oeis_id":"A257606","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A257606","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ef33085ddaeb786e","kind":"oeis_match","node_a_id":"n-c47b645304a2946e","node_b_id":null,"fingerprint":null,"oeis_id":"A049388","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A049388","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A257606","relation":"identification"},{"oeis_id":"A257606","relation":"match_reversed_triangle"},{"oeis_id":"A049388","relation":"match_row_sums"},{"oeis_id":"A257606","relation":"match_triangle"}],"bfiles":[{"node_id":"n-c47b645304a2946e","oeis_id":"A049388","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":20,"oeis_terms_available":20,"term_count":200,"raw_size":35989},{"node_id":"n-c47b645304a2946e","oeis_id":"A257606","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":36266},{"node_id":"n-c47b645304a2946e","oeis_id":"A257606","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":36266}],"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,        4;\n  2:     16,       40,        16;\n  3:     64,      296,       296,        64;\n  4:    256,     1928,      3552,      1928,        256;\n  5:   1024,    11688,     34808,     34808,      11688,       1024;\n  6:   4096,    67656,    302352,    487312,     302352,      67656,      4096;\n  7:  16384,   379240,   2423016,   5830000,    5830000,    2423016,    379240,     16384;\n  8:  65536,  2076424,  18330496,  62617144,   93280000,   62617144,  18330496,   2076424,    65536;\n  9: 262144, 11168552, 132823640, 621624968, 1309794296, 1309794296, 621624968, 132823640, 11168552, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+(n+3)*x)*P(n-1,x) + (x-x^2)*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[(4 + (n + 3)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], 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] = (4 + (n + 3)*x)*v[i - 1] + (x - x^2)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A257606","target":"triangle","first_row":0,"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:      4,        4;\n  2:     16,       40,        16;\n  3:     64,      296,       296,        64;\n  4:    256,     1928,      3552,      1928,        256;\n  5:   1024,    11688,     34808,     34808,      11688,       1024;\n  6:   4096,    67656,    302352,    487312,     302352,      67656,      4096;\n  7:  16384,   379240,   2423016,   5830000,    5830000,    2423016,    379240,     16384;\n  8:  65536,  2076424,  18330496,  62617144,   93280000,   62617144,  18330496,   2076424,    65536;\n  9: 262144, 11168552, 132823640, 621624968, 1309794296, 1309794296, 621624968, 132823640, 11168552, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+(n+3)*x)*P(n-1,x) + (x-x^2)*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[(4 + (n + 3)*x)*P[n - 1, x] + (x - x^2)*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] = (4 + (n + 3)*x)*v[i - 1] + (x - x^2)*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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A257606","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      4,        4;\n  2:     16,       40,        16;\n  3:     64,      296,       296,        64;\n  4:    256,     1928,      3552,      1928,        256;\n  5:   1024,    11688,     34808,     34808,      11688,       1024;\n  6:   4096,    67656,    302352,    487312,     302352,      67656,      4096;\n  7:  16384,   379240,   2423016,   5830000,    5830000,    2423016,    379240,     16384;\n  8:  65536,  2076424,  18330496,  62617144,   93280000,   62617144,  18330496,   2076424,    65536;\n  9: 262144, 11168552, 132823640, 621624968, 1309794296, 1309794296, 621624968, 132823640, 11168552, 262144;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (4+(n+3)*x)*P(n-1,x) + (x-x^2)*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[(4 + (n + 3)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[Reverse[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] = (4 + (n + 3)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-c47b645304a2946e","steps":[],"edges":[],"text":"A257606"},"headline":"A257606: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.","chain":null}