{"id":"n-0b0f0dd4d2c8eb05","revision":0,"first_index":0,"fingerprint":"f6-f600bd83d36e9825","name":"Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2.","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"],["1","1"],["1","9","1"],["1","59","42","1"],["1","361","925","154","1"],["1","2175","16402","8937","507","1"],["1","13061","265605","365050","67500","1587","1"],["1","78379","4127746","12611845","5592850","442242","4852","1"],["1","470289","62935117","398536866","365184855","68337922","2652742","14676","1"],["1","2821751","951081090","11977188769","20817224001","7796966547","719764976","15024830","44181","1"],["1","16930525","14314186117","349627501882","1092478534042","743589256113","135671311739","6841334432","81883843","132733","1"],["1","101583171","215034471730","10032911216685","54405946560120","63279111846004","20525571185492","2042001937491","60417569003","434638485","398430","1"],["1","609499049","3227650322541","285007169030050","2618827085889045","4992510580238064","2690455431877824","470821515539332","27621905494878","505504295171","2265229755","1195566","1"],["1","3656994319","48428773316242","8047981389614761","123262355076577975","374025758755825989","319719103004452944","91474502479131552","9405188211648986","344642965536392","4061798139602","11656124991","3587023","1"],["1","21941965941","726523024601605","226457340695486874","5715813587627918856","27027684824339496799","35452876272254259717","15772726842562580400","2628162328611005034","168939233780053326","4040062060075590","31649677474730","59446407430","10761447","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","11","103","1442","28023","712805","22857916","898132469","42280116147","2342621071428","150561531406613","11088979718611277","926332037770950778","86997458746032087751"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A166961","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","2"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169583,"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-e7a26573df71d2a3"]},{"id":169584,"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-e7a26573df71d2a3"]},{"id":169585,"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-e7a26573df71d2a3"]}],"region_exact":{},"incoming_edges":[{"id":"e-e7a26573df71d2a3","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-0b0f0dd4d2c8eb05","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 = kappa (theta + a)(theta + b) + x (u + v theta) with kappa (t + a)(t + b) = 1 + (3) t + (2) t^2 and u + v theta = -1 + 2 n + (-2) theta: The degree-local Euler step: T = kappa (theta + a)(theta + b) + x (u + v theta), theta = x D, with kappa, a, b > 0, u + v k > 0 for every k <= d and a negative comparison defect kappa (a b - (a + b + 1)^2/4) + (u - (a + b + 1) v/2) s - v^2 s^2/(4 kappa) < 0 for every s <= 0, maps every polynomial of degree <= d with nonnegative coefficients and only real roots to such a polynomial (RealRooted IsPFPolynomial.generalStep_of_natDegree_le, a half-line theorem: the symbol of T has discriminant -4 kappa s^2 times the comparison defect at s, so its real-rootedness is required for s <= 0 only, and the hypothesis (i) of P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5, for every real s, is not needed); here d = deg P_(n-1), so u(n) + v(n) k > 0 is required for 0 <= k <= deg P_(n-1) only (both ends, the condition being linear in k); by linearity a row with nonpositive coefficients is handled through its negative","lean_route":null,"lean_declarations":["RealRooted.IsPFPolynomial.generalStep_of_natDegree_le","RealRooted.EulerBidiagonal.generalStep","RealRooted.EulerBidiagonal.comparisonDefect","RealRooted.EulerBidiagonal.generalRowsDep_spec"],"certificate_hash":"ab8faf747dfcdd39c06abcc21f674510f0bc743550a80783ccd9b949ddca21fd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-13e95ce23e3c240a","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-1a6a5162ada875b1","conclusion_name":"theta_plus_one of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2."},{"id":"e-4272b05c3cda5bc7","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-59fb022e36d2469d","conclusion_name":"shift_x_plus_one of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2."},{"id":"e-5251b6f7a7c60e6c","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-bd1c359a0bd1db05","conclusion_name":"mobius_x_over_x_plus_one of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2."},{"id":"e-76091a1e3d816c6f","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-e09700269a8c9ff3","conclusion_name":"theta_plus_one_squared of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2."},{"id":"e-9960d2d1d8b1eaa0","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-9e91ddd3170293d3","conclusion_name":"reversal of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2."},{"id":"e-9ef4b91adf862752","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-521a49c8d2ed9a53","conclusion_name":"one_plus_derivative of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2."},{"id":"e-d1a2a549d583a284","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-951ab4fe319085d0","conclusion_name":"derivative of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2."}],"used_by_total":7,"links":[{"id":"l-bbc63f3ffafec884","kind":"fingerprint_related","node_a_id":"n-0b0f0dd4d2c8eb05","node_b_id":"n-9e91ddd3170293d3","fingerprint":"f6-f600bd83d36e9825","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f600bd83d36e9825"},"other_id":"n-9e91ddd3170293d3","other":{"id":"n-9e91ddd3170293d3","name":"reversal of Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2.","first_index":0,"fingerprint":"f6-f600bd83d36e9825","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","9","1"]],"card_rows":[["1"],["1","1"],["1","9","1"],["1","42","59","1"]],"row_sums_preview":["1","2","11","103","1442","28023","712805","22857916"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0b0f0dd4d2c8eb05","steps":["reverse"],"edges":["e-9960d2d1d8b1eaa0"],"text":"A166961 \u2192 reverse"},"headline":"A166961 \u2192 reverse","chain":null}},{"id":"l-8c909e6170c76e03","kind":"oeis_match","node_a_id":"n-0b0f0dd4d2c8eb05","node_b_id":null,"fingerprint":null,"oeis_id":"A166961","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":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A166961","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A166961","relation":"identification"},{"oeis_id":"A166961","relation":"match_triangle"}],"bfiles":[{"node_id":"n-0b0f0dd4d2c8eb05","oeis_id":"A166961","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":45380}],"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,       9,         1;\n  3: 1,      59,        42,           1;\n  4: 1,     361,       925,         154,           1;\n  5: 1,    2175,     16402,        8937,         507,          1;\n  6: 1,   13061,    265605,      365050,       67500,       1587,         1;\n  7: 1,   78379,   4127746,    12611845,     5592850,     442242,      4852,        1;\n  8: 1,  470289,  62935117,   398536866,   365184855,   68337922,   2652742,    14676,     1;\n  9: 1, 2821751, 951081090, 11977188769, 20817224001, 7796966547, 719764976, 15024830, 44181, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-1)*x)*P(n-1,x) + (5*x-2*x^2)*d/dx P(n-1,x) + 2*x^2*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[(1 + (2*n - 1)*x)*P[n - 1, x] + (5*x - 2*x^2)*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}]];\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] = (1 + (2*n - 1)*x)*v[i - 1] + (5*x - 2*x^2)*deriv(v[i - 1], x) + 2*x^2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A166961","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,       9,         1;\n   4: 1,      59,        42,           1;\n   5: 1,     361,       925,         154,           1;\n   6: 1,    2175,     16402,        8937,         507,          1;\n   7: 1,   13061,    265605,      365050,       67500,       1587,         1;\n   8: 1,   78379,   4127746,    12611845,     5592850,     442242,      4852,        1;\n   9: 1,  470289,  62935117,   398536866,   365184855,   68337922,   2652742,    14676,     1;\n  10: 1, 2821751, 951081090, 11977188769, 20817224001, 7796966547, 719764976, 15024830, 44181, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (5*x-2*x^2)*d/dx P(n-1,x) + 2*x^2*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[(1 + (2*n - 3)*x)*P[n - 1, x] + (5*x - 2*x^2)*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (2*n - 3)*x)*v[i - 1] + (5*x - 2*x^2)*deriv(v[i - 1], x) + 2*x^2*deriv(deriv(v[i - 1], x), x)); 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}}],"derivation":{"root":"n-0b0f0dd4d2c8eb05","steps":[],"edges":[],"text":"A166961"},"headline":"A166961: Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 2.","chain":null}