{"id":"n-8755e4112f861651","revision":0,"first_index":0,"fingerprint":"f6-c3894e4490ebd69f","name":"one_plus_derivative of basis_map_A008277 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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":false,"rows":[["1"],["4","1"],["28","12","1"],["280","160","24","1"],["3640","2520","520","40","1"],["58240","46480","11880","1280","60","1"],["1106560","987840","295960","40040","2660","84","1"],["24344320","23826880","8090880","1296960","109200","4928","112","1"],["608608000","643843200","242280640","44405760","4463760","257040","8400","144","1"],["17041024000","19280060800","7912262400","1618859200","187246080","12946080","542640","13440","180","1"],["528271744000","634002969600","280384720000","62953475200","8172472000","653304960","33023760","1053360","20460","220","1"],["17961239296000","22718375680000","10727852889600","2609663302400","373507411200","33651365440","1974255360","76211520","1912680","29920","264","1"],["664565853952000","881259515136000","441104638374400","115114384985600","17923467161600","1786864759680","118544345920","5327562240","162282120","3289000","42328","312","1"],["26582634158080000","36796205974528000","19407654326860800","5391023192755200","903746940096000","98332381347200","7239904672000","368939771200","13117104000","323443120","5405400","58240","364","1"],["1143053268797440000","1645615697037312000","910140650683264000","267394721193600000","47867129377267200","5623701244761600","453087710275200","25686893232000","1037912075200","29936146240","609929320","8550360","78260","420","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","41","465","6721","117941","2433145","57673281","1543866945","46052954821","1514472783561","54426342354385","2121878761891201","89187219264121525","4020175011403931801"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188219,"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-0c852f1bb0f4d709","e-4b5808cd01867922"]},{"id":188220,"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-0c852f1bb0f4d709","e-4b5808cd01867922"]},{"id":188221,"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-0c852f1bb0f4d709","e-4b5808cd01867922"]},{"id":188222,"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-4b5808cd01867922"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c852f1bb0f4d709","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-8755e4112f861651","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":"664a5faae89a6801a0cd7341b680abc92087c7f602c1ea5fca984db188772d72","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-b6cbf920aaa38713","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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]"}]},{"id":"e-4b5808cd01867922","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-8755e4112f861651","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"f545b15028a3826b246d8d0c8b3acfcce101c10f8df623f2148cd058136a3e85","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-a5324901f71bbafe","role":"sequence","name":"basis_map_A008277 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-3596621c5786e769","kind":"fingerprint_related","node_a_id":"n-706703e52ab62d2a","node_b_id":"n-8755e4112f861651","fingerprint":"f6-c3894e4490ebd69f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c3894e4490ebd69f"},"other_id":"n-706703e52ab62d2a","other":{"id":"n-706703e52ab62d2a","name":"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.","first_index":0,"fingerprint":"f6-c3894e4490ebd69f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","4","1"]],"card_rows":[["1"],["0","1"],["0","4","1"],["0","28","12","1"]],"row_sums_preview":["1","1","5","41","465","6721","117941","2433145"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A371080"],"derivation":{"root":"n-706703e52ab62d2a","steps":[],"edges":[],"text":"A371080"},"headline":"A371080: 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.","chain":null}},{"id":"l-e47703dd217f9391","kind":"fingerprint_related","node_a_id":"n-b0dcdb0a8a0902b9","node_b_id":"n-8755e4112f861651","fingerprint":"f6-c3894e4490ebd69f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c3894e4490ebd69f"},"other_id":"n-b0dcdb0a8a0902b9","other":{"id":"n-b0dcdb0a8a0902b9","name":"reversal 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.","first_index":0,"fingerprint":"f6-c3894e4490ebd69f","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","12","28"]],"row_sums_preview":["1","1","5","41","465","6721","117941","2433145"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-df97cd413ec843e8","steps":["reverse"],"edges":["e-83811250d0324fbe"],"text":"A035469 \u2192 reverse"},"headline":"A035469 \u2192 reverse","chain":null}},{"id":"l-f58b45d145f36f71","kind":"fingerprint_related","node_a_id":"n-df97cd413ec843e8","node_b_id":"n-8755e4112f861651","fingerprint":"f6-c3894e4490ebd69f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c3894e4490ebd69f"},"other_id":"n-df97cd413ec843e8","other":{"id":"n-df97cd413ec843e8","name":"Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.","first_index":1,"fingerprint":"f6-c3894e4490ebd69f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["28","12","1"]],"card_rows":[["1"],["4","1"],["28","12","1"],["280","160","24","1"]],"row_sums_preview":["1","5","41","465","6721","117941","2433145","57673281"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A035469"],"derivation":{"root":"n-df97cd413ec843e8","steps":[],"edges":[],"text":"A035469"},"headline":"A035469: Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.","chain":null}}],"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,           1;\n  2:          28,          12,          1;\n  3:         280,         160,         24,          1;\n  4:        3640,        2520,        520,         40,         1;\n  5:       58240,       46480,      11880,       1280,        60,        1;\n  6:     1106560,      987840,     295960,      40040,      2660,       84,      1;\n  7:    24344320,    23826880,    8090880,    1296960,    109200,     4928,    112,     1;\n  8:   608608000,   643843200,  242280640,   44405760,   4463760,   257040,   8400,   144,   1;\n  9: 17041024000, 19280060800, 7912262400, 1618859200, 187246080, 12946080, 542640, 13440, 180, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+1+x)*P(n-1,x) + 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[(3*n + 1 + x)*P[n - 1, x] + x*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] = (3*n + 1 + x)*v[i - 1] + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-64ce901fd869872f","e-0c852f1bb0f4d709"],"text":"A225470 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A225470 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}
