{"id":"n-2872b2f2cd7c5725","revision":1,"first_index":0,"fingerprint":"f6-8e0b85d88ec2ca50","name":"Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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"],["1","1"],["1","5","1"],["1","14","14","1"],["1","33","89","33","1"],["1","72","413","413","72","1"],["1","151","1632","3393","1632","151","1"],["1","310","5874","22145","22145","5874","310","1"],["1","629","19943","125456","224843","125456","19943","629","1"],["1","1268","65171","647299","1899096","1899096","647299","65171","1268","1"],["1","2547","207554","3130507","14152029","23013995","14152029","3130507","207554","2547","1"],["1","5106","649400","14455185","96451500","239047269","239047269","96451500","14455185","649400","5106","1"],["1","10225","2006913","64522294","615484672","2220047643","3369675761","2220047643","615484672","64522294","2006913","10225","1"],["1","20464","6148545","280814619","3737101485","18956099406","41587158740","41587158740","18956099406","3737101485","280814619","6148545","20464","1"],["1","40943","18721892","1199047929","21838990528","151723095958","463935053477","668764215601","463935053477","151723095958","21838990528","1199047929","18721892","40943","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","30","157","972","6961","56660","516901","5225670","57999271","701216922","9173819257","129134686520","1946194117057"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A144438","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159352,"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-d401dbe9093b268d"]},{"id":159353,"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-d401dbe9093b268d"]},{"id":159354,"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-d401dbe9093b268d"]},{"id":159355,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-d401dbe9093b268d"]}],"region_exact":{},"incoming_edges":[{"id":"e-d401dbe9093b268d","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-2872b2f2cd7c5725","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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"halfGrowth","lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"certificate_hash":"1eb8f7d97059fb56cf67294b2391c6ae5f5b72a7edcc37d6692192b599a17a9f","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":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-da103189d8330c8e","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-89920465e4889287","conclusion_name":"gamma_polynomial of Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1."},{"id":"e-39092463700c66ac","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ce79c6a77a6781f4","conclusion_name":"theta_plus_one of Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1."},{"id":"e-55abd557d63ba53c","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-d0f0d51715cde66d","conclusion_name":"theta_plus_one_squared of Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1."},{"id":"e-aa88e1a16f7cc7e2","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-b8ad7ea132013cd3","conclusion_name":"one_plus_derivative of Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1."},{"id":"e-efbd5175f6e98006","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-51731572bd1ee403","conclusion_name":"derivative of Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1."}],"used_by_total":5,"links":[{"id":"l-2dd39ddd9e7f7cc3","kind":"oeis_match","node_a_id":"n-2872b2f2cd7c5725","node_b_id":null,"fingerprint":null,"oeis_id":"A001053","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":"A001053","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9efb284f91bc7948","kind":"oeis_match","node_a_id":"n-2872b2f2cd7c5725","node_b_id":null,"fingerprint":null,"oeis_id":"A144438","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A144438","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-acbebd60ca7b4437","kind":"oeis_match","node_a_id":"n-2872b2f2cd7c5725","node_b_id":null,"fingerprint":null,"oeis_id":"A144438","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A144438","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A144438","relation":"identification"},{"oeis_id":"A144438","relation":"match_reversed_triangle"},{"oeis_id":"A001053","relation":"match_row_sums"},{"oeis_id":"A144438","relation":"match_triangle"}],"bfiles":[{"node_id":"n-2872b2f2cd7c5725","oeis_id":"A144438","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":29188},{"node_id":"n-2872b2f2cd7c5725","oeis_id":"A144438","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":29188}],"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: 1,    1;\n  2: 1,    5,     1;\n  3: 1,   14,    14,      1;\n  4: 1,   33,    89,     33,       1;\n  5: 1,   72,   413,    413,      72,       1;\n  6: 1,  151,  1632,   3393,    1632,     151,      1;\n  7: 1,  310,  5874,  22145,   22145,    5874,    310,     1;\n  8: 1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;\n  9: 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + x*v[i - 2]); 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":"A144438","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    1;\n   3: 1,    5,     1;\n   4: 1,   14,    14,      1;\n   5: 1,   33,    89,     33,       1;\n   6: 1,   72,   413,    413,      72,       1;\n   7: 1,  151,  1632,   3393,    1632,     151,      1;\n   8: 1,  310,  5874,  22145,   22145,    5874,    310,     1;\n   9: 1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;\n  10: 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + x*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A144438","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    1;\n   3: 1,    5,     1;\n   4: 1,   14,    14,      1;\n   5: 1,   33,    89,     33,       1;\n   6: 1,   72,   413,    413,      72,       1;\n   7: 1,  151,  1632,   3393,    1632,     151,      1;\n   8: 1,  310,  5874,  22145,   22145,    5874,    310,     1;\n   9: 1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;\n  10: 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + x*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-2872b2f2cd7c5725","steps":[],"edges":[],"text":"A144438"},"headline":"A144438: Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","chain":null}