{"id":"n-24ab55cfc575f41e","revision":1,"first_index":1,"fingerprint":"f6-31dfc9d837f82561","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].","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"],["2","1"],["8","8","1"],["48","72","18","1"],["384","768","288","32","1"],["3840","9600","4800","800","50","1"],["46080","138240","86400","19200","1800","72","1"],["645120","2257920","1693440","470400","58800","3528","98","1"],["10321920","41287680","36126720","12042240","1881600","150528","6272","128","1"],["185794560","836075520","836075520","325140480","60963840","6096384","338688","10368","162","1"],["3715891200","18579456000","20901888000","9289728000","2032128000","243855360","16934400","691200","16200","200","1"],["81749606400","449622835200","562028544000","281014272000","70253568000","9835499520","819624960","41817600","1306800","24200","242","1"],["1961990553600","11771943321600","16186422067200","8992456704000","2529128448000","404660551680","39341998080","2408693760","94089600","2323200","34848","288","1"],["51011754393600","331576403558400","497364605337600","303945036595200","94982823936000","17096908308480","1899656478720","135689748480","6360456960","196310400","3926208","48672","338","1"],["1428329123020800","9998303861145600","16247243774361600","10831495849574400","3723326698291200","744665339658240","93083167457280","7598625914880","415549854720","15390735360","384768384","6359808","66248","392","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","139","1473","19091","291793","5129307","101817089","2250495523","54780588561","1455367098923","41888448785857","1298019439099059","43074477771208913"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A286724","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":170020,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-01de533341b645ca"]},{"id":170021,"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-01de533341b645ca","e-aa04d2565bc4c851"]},{"id":170022,"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-01de533341b645ca","e-aa04d2565bc4c851"]},{"id":170023,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-01de533341b645ca"]},{"id":170087,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-aa04d2565bc4c851"]}],"region_exact":{},"incoming_edges":[{"id":"e-01de533341b645ca","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-24ab55cfc575f41e","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = A + B D + C D^2 with A = 2 + x, B = 4 + 4 x, C = 4 x: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]; a linear operator preserving real-rootedness preserves the alternation of two polynomials (P. Branden, Trans. Amer. Math. Soc. 358 (2006), Corollary 2.5)","lean_route":null,"lean_declarations":[],"certificate_hash":"d60deeee3dafbde0833ebe1e9e3ba97306239840998d8d6f82b30a088a7fd046","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-aa04d2565bc4c851","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-24ab55cfc575f41e","premise_count":0,"theorem":"Row-hypergeometric rows: an exact row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, proved for every n by an invariant row-annihilator system of the rule, writes every row as U_0(n) x^v (or its reversal) times the image of (1 + x)^M(n) under a product of nonnegative multiplier sequences: C^k (C > 0); k -> M'!/(M' - k)!, whose exponential generating function is (1 + x)^M' (Polya--Schur, Theorem 7.1 (3) => (1) of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"d9c822b0a70acc0de076bdbdc0ff103527bef674a2df18a8486e38db66ca12ec","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-34fd92fe8b334488","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-34454f359e0dcb07","conclusion_name":"derivative of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1]."},{"id":"e-6cd089762a4baec2","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-cadc1db6e74f2494","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1]."},{"id":"e-7d37c87889a9175f","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-c047d0120ec62185","conclusion_name":"theta_plus_one of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1]."},{"id":"e-b2ef929d973f3619","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-2e7006cec065c803","conclusion_name":"one_plus_derivative of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1]."},{"id":"e-c7d60ec169549c5b","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-f3d7765079876101","conclusion_name":"shift_x_plus_one of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1]."},{"id":"e-ea894ee43590ac57","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-17e55966d5083866","conclusion_name":"reversal of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1]."}],"used_by_total":6,"links":[{"id":"l-5fe9e6249d312ddc","kind":"fingerprint_related","node_a_id":"n-24ab55cfc575f41e","node_b_id":"n-17e55966d5083866","fingerprint":"f6-31dfc9d837f82561","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-31dfc9d837f82561"},"other_id":"n-17e55966d5083866","other":{"id":"n-17e55966d5083866","name":"reversal of Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].","first_index":1,"fingerprint":"f6-31dfc9d837f82561","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","8","8"]],"card_rows":[["1"],["1","2"],["1","8","8"],["1","18","72","48"]],"row_sums_preview":["1","3","17","139","1473","19091","291793","5129307"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-24ab55cfc575f41e","steps":["reverse"],"edges":["e-ea894ee43590ac57"],"text":"A286724 \u2192 reverse"},"headline":"A286724 \u2192 reverse","chain":null}},{"id":"l-ee2e8fa68ebe8571","kind":"fingerprint_related","node_a_id":"n-24ab55cfc575f41e","node_b_id":"n-eb866e3e5491780a","fingerprint":"f6-31dfc9d837f82561","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-31dfc9d837f82561"},"other_id":"n-eb866e3e5491780a","other":{"id":"n-eb866e3e5491780a","name":"theta_plus_one of basis_map_A008277 of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","first_index":0,"fingerprint":"f6-31dfc9d837f82561","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","24","3"]],"card_rows":[["1"],["4","2"],["24","24","3"],["192","288","72","4"]],"row_sums_preview":["1","6","51","556","7365","114546","2042551","41034456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f1c436fe49d0588b","steps":["\u03b8+1"],"edges":["e-6a06c894d6272209"],"text":"A079621 \u2192 \u03b8+1"},"headline":"A079621 \u2192 \u03b8+1","chain":null}},{"id":"l-988eb90a37126cc5","kind":"oeis_match","node_a_id":"n-24ab55cfc575f41e","node_b_id":null,"fingerprint":null,"oeis_id":"A025167","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":"A025167","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bfc15cf2575714ec","kind":"oeis_match","node_a_id":"n-24ab55cfc575f41e","node_b_id":null,"fingerprint":null,"oeis_id":"A286724","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A286724","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A286724","relation":"identification"},{"oeis_id":"A025167","relation":"match_row_sums"},{"oeis_id":"A286724","relation":"match_triangle"}],"bfiles":[{"node_id":"n-24ab55cfc575f41e","oeis_id":"A025167","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":40999},{"node_id":"n-24ab55cfc575f41e","oeis_id":"A286724","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":33124}],"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:         2,         1;\n   3:         8,         8,         1;\n   4:        48,        72,        18,         1;\n   5:       384,       768,       288,        32,        1;\n   6:      3840,      9600,      4800,       800,       50,       1;\n   7:     46080,    138240,     86400,     19200,     1800,      72,      1;\n   8:    645120,   2257920,   1693440,    470400,    58800,    3528,     98,     1;\n   9:  10321920,  41287680,  36126720,  12042240,  1881600,  150528,   6272,   128,   1;\n  10: 185794560, 836075520, 836075520, 325140480, 60963840, 6096384, 338688, 10368, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (4+4*x)*d/dx P(n-1,x) + 4*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (4 + 4*x)*D[P[n - 1, x], x] + 4*x*D[P[n - 1, x], {x, 2}]];\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] = (2 + x)*v[i - 1] + (4 + 4*x)*deriv(v[i - 1], x) + 4*x*deriv(deriv(v[i - 1], x), 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":"A286724","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:         2,         1;\n  2:         8,         8,         1;\n  3:        48,        72,        18,         1;\n  4:       384,       768,       288,        32,        1;\n  5:      3840,      9600,      4800,       800,       50,       1;\n  6:     46080,    138240,     86400,     19200,     1800,      72,      1;\n  7:    645120,   2257920,   1693440,    470400,    58800,    3528,     98,     1;\n  8:  10321920,  41287680,  36126720,  12042240,  1881600,  150528,   6272,   128,   1;\n  9: 185794560, 836075520, 836075520, 325140480, 60963840, 6096384, 338688, 10368, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (4+4*x)*d/dx P(n-1,x) + 4*x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (4 + 4*x)*D[P[n - 1, x], x] + 4*x*D[P[n - 1, x], {x, 2}]];\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] = (2 + x)*v[i - 1] + (4 + 4*x)*deriv(v[i - 1], x) + 4*x*deriv(deriv(v[i - 1], x), 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-24ab55cfc575f41e","steps":[],"edges":[],"text":"A286724"},"headline":"A286724: Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].","chain":null}